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LATENT HEATS AND VAPOUR PRESSURES
CHAPTER 3
LATENT HEATS AND VAPOUR PRESSURES Latent Heats. Definitions of latent heats for pure substances and mixtures, the Clausius-Clapeyron equation, analogues of the Clausius-Clapeyron equation applicable to mixtures, non-equilibrium latent heats, saturation specific heats. Vapour pressure equations for pure substances Wrede's equation, the Antoine equation, the Rankine-Kirchoff equation, Frost's equation. Temperature dependence of the dew- and bubble-point pressures of mixtures Accurate representation of vapour pressure and other phase boundary data using residual plots Use of residual plots for calculating latent heats from vapour pressure data Comparison of experimental and calculated latent heats Determination of second virial coefficients for vapours from latent heat and vapour pressure data. Extension of vapour pressure data
LATENT HEATS
The molar latent heat of a pure substance is defined as the heat required to convert one mole of that substance from the liquid to the vapour state under equilibrium conditions at constant temperature and hence constant pressure. Thus
S is the entropy of a closed system containing nG moles of gas phase. In the case of a mixture, the pressure and composition may change con tinuously during an isothermal evaporation and likewise the temperature and composition may change continuously during an isobaric evaporation process. In this case the heat required per mole of liquid converted may change continuously during the evaporation process. Suppose that one mole of a liquid mixture of specified composition is enclosed in a container and brought to the bubble point. The isothermal integral latent heat LT is defined as the heat required to convert the liquid completely to the vapour phase at constant temperature. Similarly the isobaric integral latent heat LF is the heat required to convert the liquid to the vapour phase at constant pressure. The differential molar latent heats at constant temperature and 175 13 PE (AP—250)
176
PHASE EQUILIBRIUM IN MIXTURES
constant pressure are defined by the equations
and
dS 1 dS \ / = T-= T- -—c\ T dnG\T dMGjTX
(3.1a) '
K
respectively. A/ G is the fraction of the moles of mixture which are present in the vapour phase and 5 is the molar entropy of the system. The isothermal integral and isobaric integral molar latent heats may be obtained from the differential latent heats using the equations I
LT=[lTdMG]TX
(3.2a)
LP = $lPdMG]PX
(3.2b)
and 6
lT and lP may be calculated from phase boundary data using equations (9.44) and (9.53) respectively. These equations are somewhat complex to use and detailed consideration of them will be deferred until Chapter 9. The isothermal integral latent heats of a series of mixtures of propylene and carbon dioxide are shown in Fig. 3*1.
THE CLAUSIUS-CLAPEYRON EQUATION FOR THE LATENT HEAT L OF A PURE SUBSTANCE
Consider a system containing one component only and consisting of a vapour phase in equilibrium with a liquid phase at pressure p° and tem perature T. For equilibrium, the chemical potential of this component, and hence the molar Gibbs function, must be the same in the two phases. The molar Gibbs function of any phase in a one-component system is a function only of the pressure and temperature, so for any infinitesimal change in the state of the phase dG
L-£L
dp = 4?"l ^rl dr dP )T + "T dTjp
= VdP-SdT
(3.3)
LATENT HEATS AND VAPOUR PRESSURES
177
Suppose that the temperature of the two-phase system changes from T to T + dT and the vapour pressure changes from p° to p° + dp°. The cor responding changes in the molar Gibbs function of the vapour and liquid phases are dGG and dGL and, since G° = GL before and after the change, dGG = dGL
Fio. 3*1. The isothermal integral latent heats of mixtures of carbon dioxide and propylene.
Combining this result with equation (3.3) Vcdp° - SGdT = FLd/?° - 5 L d T or L = T(SG - SL)=T(VG
- VL)^-
(3.4)
This is the well-known Clausius-Clapeyron equation. Provided care is taken in evaluating the slope dp°ldT of the vapour pressure line, this equa tion may be used to calculate latent heats with a precision equal to that attained by calorimetric methods. The determination of this slope is dis cussed further on pp. 189 — 95.
13*
178
PHASE EQUILIBRIUM IN MIXTURES
EQUATION FOR THE ISOTHERMAL INTEGRAL LATENT HEAT OF A MIXTURE
The isothermal integral latent heat may be calculated using the equation1 dP T
D.P.
dT)x
-v
B.P. ] B.P. dP +
dr
W([AG]X)X
(3.5)
i ^O^A, j
N.
N. S
If FIG. 3*2. Illustrating the derivation of equation (3.5) for the isothermal integral latent heat of a mixture.
[AG]X is the isothermal Gibbs function of condensation and is defined by the equation pu.r.
[AG}X=PS
VdP]TX
(3.6)
pD.P.
For a pure substance, P D P - = P a P - hence [AG]X = 0. F D R is the molar volume of the mixture at the dew point at the temperature T. VBF' is the molar volume at the bubble point. dP/dTf**' and dP/dT)BV- are respec tively the rate of change of the dew-point pressure and of the bubble-point pressure with temperature for a mixture of overall composition x. In the case of a pure substance, [AG]X = 0, dP/dT)^ = dP/d7)*p* = dp0/dT, and equation (3.5) reduces to the simple Clausius-Clapeyron equation L / r = (K DP - - VBJ,)dp°/dT
= {VG - VL)
dT
The validity of equation (3.5) may be demonstrated graphically as in Fig. 3*2. The full lines in this figure show the variation with pressure of the volume 1
HASELDEN, HOLLAND, KING and STRICKLAND-CONSTABLE, Proc. Roy, Soc. A 240,
13 (1957).
179
LATENT HEATS AND VAPOUR PRESSURES
of an arbitrary mixture in the two-phase region at some temperature T; the dotted lines show the state of affairs at a slightly higher temperature T + dr. The increase in temperature from T to T + dT is accompanied by changes not only in the volume at any given pressure but also in the values of the terminal pressures P DP - and P B P \ It is desired to determine the
FIG. 3*3. Illustrating the determination of the gradients required when using equation (3.5) to evaluate the isothermal latent heat of a mixture, (a) di7dr) DP - and djP/dr)BP- are obtained from thePTdiagram for the mixture in question, (b) The overall molar volume for the system is plotted as a funcpB.P.
tion of pressure at each of a series of temperatures and [A G]x = J VdP] is pD.P
obtained at each temperature by graphical integration, (c) d[A G]x/dT obtained from a plot of [AG]X as a function of temperature.
B.P.
difference d( J VdP) between the Gibbs function of condensation at the D.P.
higher temperature (which is the area enclosed between the dotted lines) and the Gibbs function of condensation at the lower temperature (which is the area enclosed between the full lines in Fig. 3*2). It is seen from the figure that the difference between these areas is equal to Ax+ A2- Az FB.P.
jpB.P.
+
|BJ' d K | D.P. dT
dp
j dT
i.e. to _
F D.P. dj pD.l
180
PHASE EQUILIBRIUM IN MIXTURES
where dP BP * and dPD,p* are respectively the increase in the pressure at the bubble point and the increase in pressure at the dew point when the temperature is increased from T to T + dT. Dividing by AT, it follows that B.P
dP)BP'
d/MDP-
D.P.
a i
Q1
IX
JX
BP
-
D.P.
dP. p
Since j 1
B.P.
Ay\
D.P.
QI
)P
1
B.P.
-i
\X
D.P.
ATX
the equation
follows at once. Provided that accurate volumetric data for the mixture are available within the two-phase region, equation (3.5) probably provides the most accurate method of calculating integral isothermal latent heats. The gradientsdP/dr) DR and dP/dr) B ' P, may be obtained from the PTdiagram for the mixture in question (Fig. 3*3a). The quantity [^G]^ may be obtained, at each of a set of temperatures, from equation (3.1), using a graphical technique to obtain the integral in each case (Fig. 3'3b). For this purpose, values of the overall volume of the mixture are required as a function of pressure at each temperature. Such data could in principle be obtained by enclosing a sample of the mixture in question in the apparatus shown in Fig. 2*1, and directly noting the total volume as a function of pressure. In order to evaluate the gradients dPDF/dT)x, dPBF/dT)x and d[AG]x/dT accurately, it is desirable to use residual plots. A worked example of such a calculation is given on p. 195, and the relative magnitudes of the terms
dT)x
dT)x
dT
for this case are shown. OTHER ANALOGUES OF THE CLAUSIUS-CLAPEYRON EQUATION APPLICABLE TO MIXTURES
A variety of equations may be used to calculate the latent heats of mix tures and all of these could be described as analogues of the ClausiusClapeyron equation. Equations (9.44) and (9.53) have already been men tioned. In addition, equations (9.30) and (9.41) have the same form as the Clausius-Clapeyron equation and may be used to calculate liquid phase
LATENT HEATS AND VAPOUR PRESSURES
181
enthalpies from those of the equilibrium vapour phase. Equation (3.6) differs from the above in that the data required when using it may be derived entirely from volumetric observations on the mixture for which the latent beat is required. No data for neighbouring mixtures are needed and a knowledge of the composition of the equilibrium gas and liquid phases is also unnecessary. SPECIFIC LATENT HEATS
The equations so far considered are for the molar latent heat. Analogous equations exist for the latent heat per unit weight of distillate. These may be obtained by substituting V(w) for V, x(w) for x, in the above equations. Thus equation (3.5) for the latent heat of a pure substance becomes L(w) = r [ r G ( w ) - K L ( v v ) ) - ^ ) and so on. (L(w) is the heat required to convert unit mass of substance from the vapour to the liquid state. VG(w) is the volume occupied by unit mass of vapour and VL(w) the volume occupied by unit mass of liquid at the temperature T and the vapour pressure p°. NON-EQUILIBRIUM LATENT HEATS
The latent heats considered above all refer to a phase change carried out in such a way that only infinitesimal displacements from equilibrium con ditions are allowed. Such conditions are never exactly satisfied. In practice, such deviations do not have an appreciable effect on the latent heat, except in the case of substances such as hydrogen fluoride which polymerize strongly in the gas phase, the degree of polymerization varying rapidly with the pressure.2 The heat of polymerization of hydrogen fluoride is large and it has been found that the non-equilibrium latent heat of vaporization at any given temperature does depend noticeably on the pressure at which evaporation is carried out. SATURATION SPECIFIC HEATS
The saturation specific heat a° for a vapour is the heat required to raise the temperature of unit quantity of vapour by unit amount, the pressure being adjusted so as to maintain the vapour at its dew point. Thus
2
SIMONS and HILDEBRAND, / . Am, Chem. Soc. 46, 2183 (1924).
182
PHASE EQUILIBRIUM IN MIXTURES
Similarly the saturation specific heat GL for a liquid is the heat required to raise the temperature of unit quantity of liquid by unit amount, the pressure being adjusted to maintain the liquid at its bubble point. Thus dS \BR
T
Since
(3 8)
•*- Hrl dS ;D-P- _ j)S_\ drj,
" dT)PX
+
-
J5S \
dP \DP»
dP)„
AT)X
'
it follows on combining (3.7) with (1.35) that 8VG\
dP lDJP-
c® is the constant pressure specific heat of the gas phase and the gradient dVG/dT)PX is the rate of change of volume with temperature at constant pressure in the gas phase, the gradient being taken at the dew point. Similarly, T JP dT) •"-*- r Tr),dfJ
<3 10>
-
Now
Lr=r(SDP-SBP) (3.11) D,P where LT is the isothermal latent heat, S - is the entropy of the system at the dew point and SBP* is the entropy of the system of the same com position at the same temperature, but at the bubble-point pressure. Dividing (3.11) by T, differentiating with respect to temperature and combining with (3.7) and (3.8),
Equations (3.7) to (3.12) are equally applicable to a pure substance or a mixture. VAPOUR PRESSURE EQUATIONS FOR PURE SUBSTANCES
The oldest and simplest equation for representing the vapour pressure of a pure substance as a function of temperature is due to Wrede.8 According to this equation, \og10p°=a-blT (3.13) *Ann.Phys. 53, 225(1841).
LATENT HEATS AND VAPOUR PRESSURES
183
Figure 3*5a on p. 192 shows the logarithm of the vapour pressure of propylene plotted as a function of the reciprocal of the absolute temperature from the normal boiling point to the critical point. It is found that throughout this extensive range the Wrede relationship represents the data to within ± 3 per cent. Although not rigidly accurate, the Wrede equation is, because of its simplicity, very useful as a "first order" equation and it is suitable for use in conjunction with a residual plot (see p. 189). Numerous other equations are in use for representing vapour pressure data for pure substances. The following may be mentioned. l°gio/>0== A -
B
KC + 0 (Antoine equation)
(3.14)
'ogiop°= c - d/T + elog T (Rankine-Kirchhoff equation) (3.15) log10 p° = f-
9IT - h log T + jP/T2 (Frost's equation)
(3.16)
The extent to which these and other equations in fact represent the experi mental data has been discussed by Waring.4 The Antoine and RankineKirchhoff equations are successful over a rather larger range of conditions than the Wrede equation but a four-parameter equation such as (3.16) is required to fit vapour pressure data satisfactorily from the triple point up to the critical point. The Wrede equation may be derived by integrating the Gausius-Clapeyron equation L = T(VG - VL) d(p°)/6T (3.4) if it is assumed that the latent heat L is constant, that the volume VG of the gas phase is given by the perfect gas law and that the volume F L of the liquid phase is negligible compared with VG. Putting VG = RT/p° =^ VG — FL, equation (3.4) becomes LdT _ Rdp°
~W
p°
Integrating for constant L, ]np°= -L/RT+
C
(3.17)
This is Wrede's equation with a = C/2-303 and b = L/2-303 R. All three of the assumptions upon which the above derivation is based break down completely as the critical point is approached. The approximate validity of the equation in the critical region must therefore be regarded as for* Ind. Eng. Chem. 46, 762 (1954).
184
PHASE EQUILIBRIUM IN MIXTURES
tuitous. In this connection, it is of interest to note that the ClausiusClapeyron equation may be rewritten in the form AT where Z G and Z L are the compressibility factors of the gas and liquid phases. Hence
d/>° p°
=
L
G
L
R{Z - Z )
G L J_ = _ (// -// )
T
1
1
(3.18)
"^(z^^z )" y
Provided the ratio L/(ZG — ZL) is independent of temperature, equation (3.18) may be integrated to give the Wrede equation, with b = L/2-303 R(ZG - ZL). The fact that Wrede's equation remains approxi mately true even up to the critical temperature indicates that the ratio L/(ZG — ZL) is approximately constant for a given substance. The Wrede equation represents vapour pressure data for solids quite adequately. Table 3.1 shows the constants a and b for a few substances in the solid state. The pressure range over which the equation is applicable and the precision with which the equation represents the experimental vapour TABLE 3.1.
CONSTANTS IN THE W R E D E EQUATION
(p° in m m of mercury, T in °K) Range (mm)
Substance Phenol tf-Cresol p-Cresol Acetylene Ammonia a
11-5638 12-7778 120298 8-8400 100059
3586-36 3970-17 3861-98 1127-1 1630-7
Precision (mm)
0-03-1-04 0-02-0-40 001-0-29 up to 962 7-45
+ 0-03 a ± 001a ± 0-009° _b
± 2 percent
BIDDISCOMBE and MARTIN, Trans. Faraday Soc. 54, 1316 (1958).
* BURRELL and ROBERTSON, / . Am. Chem. Soc. 37, 2188 (1915). e GIAUQUE and OVERSTREET, / . Am. Chem. Soc. 59, 254 (1937).
pressures are also shown. The Antoine equation* (3.14) is frequently used to represent vapour pressure data at low reduced pressures (up to say P r = 0-1). In this range it is rather more accurate than the simple Wrede equation. The constants may be derived by a method described by Willingham et al.5 The constants A, B and C are shown for various liquids in Table 3.2. * ANTOINE, Compt. rend. 107, 681 (1945). WILLINOHAM et al., J. Research Nat. Bur. Standards 35, 219 (1945).
6
185
LATENT HEATS AND VAPOUR PRESSURES TABLE 3.2.
CONSTANTS IN THE ANTOINE EQUATION
(p° in mm of mercury, t in °C) Substance Ethanol n-Propanol Triethylamine Phenol 0-Cresol /7-Cresol n-Pentane n-Hexane n-Decane Benzene Carbon tetrachloride
A 8-11219 7-9967 6-9952 7-13457 7-07055 7-11767 6-87372 6-87776 6-95367 1 6-89324 6-85906
Range (mm)
B
C
159219 1584-41 1300-30 1516-072 1542-299 1566029 1075-816 1171-530 1501-268 1203-835
226062 212-47 226-50 174-569 177-110 167-680 233-359 224-366 194-480 219-924
100-770 150-790 56-781 352-780 88-780 57-780 57-780
1196-60
224-138
90-770
90-770
Precision ± ± ± ± ± ± ± ± ± ±
0-16 mm' 0-03°C* 0-03°C* 0-007°C*«c 0-006°C*-c 0006°C**C 0-23 mm" 0-41 mmrf 0-25 mm* 0-24 mm
+ 0-21 mm
* Worst values.
a
BARKER, BROWN and SMITH, Disc. Faraday Soc. No 15, 143 (1953).
*COPP and FINDLAY, Trans. Faraday Soc. 56, 13 (1960). C BIDDISCOMBE and MARTIN, Trans. Faraday Soc. 54, 1316 (1958). d Ref. 5.
The Rankine-Kirchhoff equation (3*15) also represents vapour pressures of pure substances rather more accurately than Wrede's equation. Equation (3.15) may be deduced from the Clausius-Clapeyron equation, if a linear variation of the latent heat with temperature is assumed. The assumptions of an ideal gas phase and a liquid phase of negligible volume are also made as in the derivation of equation (3.17). It is convenient to write the latent heat in the form L = T(SG - SL) = (HG - HL) where HG is the molar enthalpy of the gas phase and / / L is the molar enthalpy of the equilibrium liquid. Differentiating the above expression for L with respect to tem perature, dL dT
d// G dT
dHL dT
It should be remembered that the gradients di/°/drand dHL/dT are not taken at constant pressure but along the saturation line. Thus dHG/dT is the rate at which the enthalpy of the saturated vapour phase changes with temperature. Now di/ G =
dH
err,
dinG
(
186
PHASE EQUILIBRIUM IN MIXTURES
and
dH\L
dHL
7 f I,
Hence dL ~dT
C
~ P-
_ dH iL dr + dH\c
C
P + \jp)T
8H\L]
-
TpjJ'dT
(3.19)
Except in the close vicinity of the critical point, the term(dHL/dP)T may be neglected and, if the gas phase is perfect, (dH/dP)T = 0. Under these conditions therefore, equation (3.19) reduces to dL 7 T
B
<119a)
^
where A cp is the difference between the constant pressure specific heats of the vapour and liquid phases. If it is assumed that A cp is independent of temperature, (3.19a) may be integrated to give L = L° + A cpT
(3.20)
The constant L° may be regarded as being the latent heat which the liquid would have if it could be subcooled to the absolute zero of temperature. Combining equations (3.20) and (3.4) and setting (V°-VL) = (RT/P°\ the equation dp° _ LpdT AcpdT p° "" RT2 + RT is obtained. Hence, on integrating,
y
RT
R
or
This expression for the vapour pressure is identical with equation (3.15), with d = (L°/2-303 R) and e=(AcpjR). At temperatures removed from the critical temperature, equation (3.15) is sufficiently accurate to summarize most available vapour pressure data within the experimental error. A table, giving the constants c, d and e for a number of liquids, has been given by
LATENT HHATS AND VAPOUR PRESSURES
187
Moelwyn-Hughes.6 It is found that the constant e is normally negative and that its absolute value increases with the molecular complexity. The constant d0 likewise increases with the size of the molecule, as is to be expected since it is directly proportional to L°, the latent heat extrapolated to a limiting low temperature. A further discussion of the dependence of the parameters c, d and e on the molecular structure is given by MoelwynHughes.7 Compared with the Antoine equation, the Kirchhoff equation suffers from the disadvantage that it is not possible to use it to calculate explicitly the boiling point temperature as a function of pressure. However, the Kirchhoff equation has the advantage that the constants in this equation have a physical interpretation. Neither the Kirchhoff nor the Antoine equation is suitable for use in the critical region. The Frost—Kalkwarf equation (3.16) may be used throughout the range from the triple point to the critical point. The equation was originally derived8 by integrating the Clausius-Clapeyron equation with the assump tions (a) that L is given by (3.20) with A c independent of temperature, (b) that V° — VL is correctly given by van der Waal's equation. However, the equation is probably best regarded as empirical. The first three terms are identical with those of the Kirchhoff equation; the last term is small at low reduced pressures but becomes large as the critical point is ap proached. In order to use the equation to calculate the vapour pressure it is necessary to employ an iterative method. One possible procedure is first to calculate the vapour pressure using the first three terms only; the value Px thus obtained may then be substituted into the last term and a new value P2 is calculated from the equation log P*=f-glT-
h log T + JPJT2
The above process is repeated until the difference (Px — P2) is negligibly small. Perry and Thodos9 have evaluated the constants f9g, h,j for the normal alkanes from methane to n-dodecane and have represented these constants as functions of the number of carbon atoms in the alkane chain. They obtain an average deviation in the calculated pressure of ±0-27 per cent. The constants/, g, h9j for a few hydrocarbons are shown in Table 3.3. The values for the unsaturated hydrocarbons are taken from a paper by Smith and Thodos.10 These authors list vapour pressure constants for many other unsaturated hydrocarbons and suggest a method by which the values of the constants may be estimated from a knowledge of molecular structure and the normal boiling point. 6
Physical Chemistry, pp. 699, 701, 2nd edition, Pergamon Press, 1965. Ref. 6, p. 696. 8 FROST and KALKWARF, / . Chem. Phys. 21, 264 (1953). 9 Ind. Eng. Chem. 44, 1649 (1952). 10 A. I. Ch. E. Journal 6, 569 (1960). 7
188
PHASE EQUILIBRIUM IN MIXTURES TABLE 3.3.
CONSTANTS IN THE FROST EQUATION
(p° in mm of mercury, T in °K) Substance Methane Ethane Propane n-Heptane n-Octane Ethylene Propylene 1-Heptene 1-Octene Acetylene Propyne
/
9
14-85689 16-51824 18-17125 28-10450 31-52935 1413307 1913914 25-90236 27-94965 18-37039 22-94034
592-41 1061-2 1397-4 2813-7 3215-5 997-69 1409-59 2659-36 2989-80 1188-90 1759-46
// 3-26150 3-48408 3-91480 6-87452 7-9220 3-75709 4-26637 6-15555 6-74267 403099 5-43367
J 0-23785 0-45265 0-70362 2-8875 3-8008 0-38247 0-70426 2-47607 303649 0-37407 0-43472
Example Use the Antoine equation to calculate the normal boiling point of phenol and also the latent heat of phenol at its normal boiling point. (See Table 3.2 and mention any assumptions which you make in your calculation.) (Answer: L = 16410 cal/mole) TEMPERATURE DEPENDENCE OF THE DEW- AND BUBBLE-POINT PRESSURES OF MIXTURES
Working over a sufficiently limited pressure range, it is possible to repre sent the dew- and bubble-point pressures and also the relative volatility for a mixture of fixed overall composition by the Wrede-type equations11 \ogiQP™=A-BIT]x log 10 / ,ap - = C-D/T]X log10a=E + F/T]x
(3.21) (3.22) (3.23)
The parameters A, B, C, D, may be plotted as functions of composition. This provides a convenient method of smoothing dew- and bubble-point data and also of interpolating the dew- and bubble-point pressures at com positions other than those for which they were determined experimentally. Unfortunately, as seen in the previous section, these equations are only valid over a rather limited range of conditions. For this reason, it is desirable to use the above equations in conjunction with residual plots as described in the next section. 11 DIN, Trans. Faraday So:. 56, 668 (1960). (System N 2 - 0 2 over pressure range 1-10 atm.)
LATENT HEATS AND VAPOUR PRESSURES
189
ACCURATE REPRESENTATION OF VAPOUR PRESSURE AND OTHER PHASE BOUNDARY DATA USING RESIDUAL PLOTS
If the primary vapour-liquid equilibrium data are to be used for sub sequent calculations or interpolations or even if they are to be checked for errors, it is necessary to find some means, analytical or graphical, of repre senting the data to the full degree of accuracy with which they were obtained.
/ /
(a)
i
X) 55
X
FIG. 3*4. Illustrating the construction of a residual plot. The purely analytical method is tedious, while the use of a direct graphical plot is frequently unsatisfactory since the data are usually obtained with an accuracy greater than that with which they can be represented on a piece of graph paper of reasonable size. Furthermore, even if a sufficiently large piece of graph paper is used, the spacing of the points may be large com pared with the curvature of the curves on which they lie, so that it is difficult to draw a best curve through the points or to see whether a point is "off line". A good example of a direct plot is given by Fig. 2*13, which shows the dew-point and bubble-point lines for mixtures of the system C 3 H 6 -C0 2 as a function of temperature. The experimental accuracy of the pressure measurements is claimed to be about ±0-01 atmosphere, while that of the temperature measurements is given as ±0-05°C. It is impossible, however, using the direct plot, to tell whether or not a given point actually lies on the curve to within these limits. The most satisfactory way of manipulating such data is to use the method of residual plots, which may be regarded as
190
PHASE EQUILIBRIUM IN MIXTURES
a combination of the analytical and graphical methods. As a first stage in using this method, an equation is found which approximately represents the experimental data (say of y as a function of x). It is convenient to call this equation the "first order equation". The small difference between the experimental and "calculated" values of the variable y is called the "residual" value of y and is plotted graphically as a function of x. This graph is called the residual plot. Examples of such plots are shown in Figs. 3*4, 3*5, 3*7 and 3*8. If a satisfactory form for the first order equation has been found, there should be no sharp points of inflexion and it should be possible to observe the experimental scatter on the residual plot. Since by definition, .Vres'dual
(x) (3.24) it is possible to interpolate a value for y at any given value for x. yca]c (x) is obtained from the first order equation and JVcsiduaiC*) *s r ^ a d from the residual plot. Gradients may similarly be determined more accurately than from the direct plot of y as a function of x. For this purpose, the equation
f -IfI +lf) ax
\ ax jcalc
\ ax jreSjdUai
<3-25)
is used. (dyldx)ca{c is obtained by differentiating the first order equation. (dj>/dx)reSidual may be determined from the residual plot. The error in this determination of (d>>/d;t)residual is unlikely to be less than ±5 per cent. However, since the magnitude of j>reSiduai s n o u ld normally be less than 1/10 that of y9 the error in the determination of dy/dx may be less than 0-5 per cent. The accuracy with which dy/dx may be determined by this method in any particular case depends on the accuracy of the data and on the extent to which these data are represented by the first order equation. When calculating (d}VdA')residuai, a tangent must be drawn to the residual curve. To facilitate this, a front-silvered (and preferably half-silvered) mirror and set-square may sometimes be required. The mirror is placed with its reflecting surface vertical to the plane of the graph paper and this surface is made to cut the curve at the point A at which the gradient is required. The mirror is now rotated about a vertical axis through A until no discontinuity can be seen between the gradient of the curve and its reflection at the point A. When this has been achieved, the actual curve and its reflection in the mirror should appear to lie on one continuous curve. The base of the mirror will then be perpendicular to a tangent drawn to the curve at the point A. The desired gradient may be determined using a transparent straight edge permanently attached to the back surface of the mirror. The mirror may be prepared from a flat glass plate, (a 2*/4 in. x 3V4 in. photographic plate with the emulsion removed is quite suitable) using the method of Twyman.12 Any deposit on the back side of the mirror 12
Trans. Optical Soc. 24, 203 (1922-3).
191
LATENT HEATS AND VAPOUR PRESSURES TABLE 3.4.
ANALYTICAL EQUATIONS WHICH APPROXIMATELY REPRESENT PHASE
EQUILIBRIUM DATA AND WHICH ARE SUITABLE FOR USE IN CONJUNCTION WITH A RESIDUAL PLOT
Variables
Quantities to be maintained constant
Equation
Dew-point, pressure and temperature
Composition
io gl0 P D - p - = . 4 - 2 ? / r A
Bubble-point pressure and temperature
Composition
Isothermal latent heat and temperature of vaporiza tion
log10i>BP-
Composition
Isothermal Gibbs function of condensation and tem perature
log 10 £r = £log 1 0 (Tc-T)+
Composition
Bubble-point pressure and composition
log 1 0 JG = / l o g
Temperature
=C-DjT]x
1 0
F
(r-D+/
i
should be removed. A beading placed along three sides of the mirror will protect the silvered surface. It should be noted however that in most cases the greatest contribution to the uncertainty in (dj/dx)ieS{dual will arise from the error involved in drawing the residual curve through the experimental points rather than from any error in determining the gradient of the curve as drawn. In these cases there is little point in using the mirror method to determine the gradient. Where possible, it is convenient to write the first order equation in the form
to) = ™/2(*) + c
(3.26)
where y is one of the experimental variables (say pressure or volume) and x is the other variable (say composition or temperature)./! and/ 2 are simple analytical functions such as log10 or reciprocal. The form of the functions / i and/ 2 may already be well known, otherwise intelligent guess-work must be used. Table 3.4 shows suitable forms for a number of cases. In order to see whether the guess for the form of/x and/ 2 is a good one, the experi mental values of x and y are used to calculate corresponding values of fx(x) and f2(y), and/i(x:) is then plotted against f2{y). If the first order equa tion (3.26) is a satisfactory one, this plot should be approximately linear, and should not show any sharp points of inflexion. If such a plot has been obtained, a straight line is drawn through the points, the gradient and intercept of this line providing the quantities m and c respectively in 14 AP-250
192
PHASE EQUILIBRIUM IN MIXTURES
(a)
2
n
1
3
r
Pressure, atm 10
15
20
30
40
-$>C-
\
£
X
B -04
./
\
-60
-40
-20
0 20 Temperature,
40 °C
(b)
t
r
60
80
FIG. 3-5. The vapour pressure of propylene.
LATENT HEATS AND VAPOUR PRESSURES
193
the first order equation. This may now be written down explicitly. As an example of the use of the method of residual plots, further consideration will now be given to the dew- and bubble-point pressures for mixtures of carbon dioxide and propylene. It is known that the vapour pressure of a pure substance is represented quite closely as a function of temperature by the equation log10/>° =a-b/T (3.13) When considering dew- and bubble-point pressures of a mixture, it would therefore seem reasonable, as afirstguess, to try the same form of equation. Figure 3"5a shows the logarithm of the vapour pressure of pure propylene
I
i
i
i
30
31
3-2
3-3
i
i
3-4 3-5 t/T, °K-'
i
i
i
l
3-6
3-7
3-8
3-9
FIG. 3*6. The logarithms of the dew-point and bubble-point pressures of a 50 mole per cent mixture of carbon dioxide and propylene shown as functions of the reciprocal of the absolute temperature.
plotted as a function of the reciprocal of the absolute temperature, and Fig. 3*6 shows the logarithms of the dew point pressure and of the bubblepoint pressure of a mixture of carbon dioxide and propylene containing 49-7 mole per cent C0 2 plotted as functions of the reciprocal of the absolute temperature. It is seen that except in the immediate vicinity of the critical point a nearly straight line relationship is obtained in each case. A similar result is obtained for the remaining mixtures indicating that equations (3.13), (3.21) and (3.22) are satisfactory for use as "first order" equations over a wide range of conditions. Values for the constants in these equations may conveniently be evaluated from the experimental points at the ex tremes of the range considered. For example, it is known that the bubblepoint pressure for the mixture containing 49-7 mole per cent carbon dioxide is 14-81 atm at -15°C and 57-98 atm at 50°C So log10PBP- = 1-1706 for 14*
194
PHASE EQUILIBRIUM IN MIXTURES
l/r = (3-8737 x l O - y K - 1 and log 10 P BP ' = 1-7633 for 1/r = (3-0944 x x lO-yK- 1 . Hence suitable values to chose for D and C in equation (3.22) 1-7633-11706 = 0-7606x3-0944 and = 3-8737-3-0944 = 4-1169. In a similar way it is found that the equation log10 p° = 4-3380 -978-0/rshould be suitable for use as afirstorder equation when represent ing the vapour pressure of pure propylene and that the equation log10 PUFt = 4-7648 - 1014-5/r should be a satisfactory first order equation to use when representing the dew-point pressure of the above mixture as a function
__! -10
1 0
I 10
I 20 Temperature,
I 30
1 40
I 50
I 60
°C
FIG 3*7. Residual plots giving the dew-point and bubble-point pressures of a 50 mole per cent mixture of carbon dioxide and propylene as functions of temperature. These plots are to be used in conjunction with the equations log 10 P u? ' = 4-7648-1014-5/r and log10/>B*p- = 4-1169- 760-6/r where P is expressed in atm and Tin °K.
of temperature. Residual plots based on these equations are shown in Fig. 3-7. The experimental uncertainty in the pressure is estimated to be ±0-01 atm and this uncertainty is indicated by a vertical line through each point on the residual plots. No great care is required in the evaluation of the constants in thefirstorder equation, since it is sufficient that the equation should approximately represent the experimental data.
195
LATENT HEATS AND VAPOUR PRESSURES ILLUSTRATIONS OF THE USE OF RESIDUAL PLOTS IN LATENT HEAT CALCULATIONS
Illustration 1 Calculate the latent heat of pure propylene at 30°C, given that the volumes of the saturated gas and liquid phases at this temperature are respectively 1-5233 and 0-0858 l./mole. From equation (3.4), OLv) L=T(VG-V
L v
d
^
dT
Now dp°
d/>°calc
d/£esidual
dT
dT
dT
where
dT
( /&lc ) I d l o g /?calc -[-%■][ \ T }{ d(l/T) 10
2
x 2-303
Using the equation logl0 p° = 4-3378 — 978-0/J and the corresponding residual plot, it follows that at 30°C
Jt
dr
A* 7 1 O ^
IV OOl
S\ XV
AK
A J U J —• v J I / 1
an
OT/-1
and that Residual
dT
= - 0 0 0 7 8 ± 0008 atm °K- 1
Hence ^30°C
= 1-4375 x 0-3092 = 134-75 ± 0*4 1., atm/mole = 779 ± 2 joules,toole Illustration 2
Calculate the isothermal latent heat of condensation of a 50 mole per cent mixture of propylene and carbon dioxide at 30°C, given that the molar volumes of the mixture at the dew-point and bubble-point pressures are 0-7185 and 0-0754 l./mole respectively. The isothermal Gibbs function of condensation for this mixture has been evaluated by graphical integration
196
PHASE EQUILIBRIUM IN MIXTURES
(see Fig. 3#3b). It is found to vary with temperature as follows
t°C 15 0 15 30 40 50
[AG]X (1. atm/mole) 8-174 7-197 6-291 5-287 4-458 3-377
The critical or "plait point" temperature for the mixture is known to be 60-4°C. Method From equation (3.5), T
dT)x
AT)X
The term I/D.P.
dr
U
lxJX
d/> fDPdT
may be written equal to i/D.P. I
dpD.P. calc
KXA
dT
,
dp®?UA residual
dT
where P^ic* *s gi v e n by the equation logio^aic* = 476478 - (1-01445 x 10 3 )/r and Pje'sTciuai *s shown plotted as a function of temperature in Fig. 3*7. dPcDaicp/dr = p ° £ d io g e pcDalcp/dr 1-01445 = 2-3026 (antilog (476478 - 101445/7)) x — - ^ ~ so at 30° dPcDalp/dr = 2-3026 x26-21 2 x 1-01445 x 10-881 = 0-6662 atm "K-1. From Fig. 37, dP%MUJdT = -0-0108 ± -001 atm °K-\ Hence FD-p-
j p
F BP - dP/dT)V- = 0-0754 W~r- + —
r
\
^ L
LATENT HEATS AND VAPOUR PRESSURES
197
where log10P^te = 4-11685 — 0-76058/r and P^duai i s shown as a func tion of temperature in Fig. 3*7. From this plot it is found that dPf^^JdT = 0-0054 atm 0 K~ 1 at 30°C Hence V**dPldT)y- = 0-0754(0-3357 x 2303 + 00054) = 00587 ± 00001 (1. atm)/(mole °K) In order to evaluate the quantity (dldT)([AG]x), it is desirable to obtain a suitable first order equation representing the variation of [AG]X with
-20
-10
10
20
Temperature,
30 °C
FIG. 3*8. The residual isothermal Gibbs function of condensation for a 50 mole per cent mixture of carbon dioxide and propylene. This graph is to be used in conjunction with the equations [AGh^AG^ and
+ A (7rcsldual
log10 AG^ = 0-4462 log10 (333-6-T) + 0*0747
temperature. A plot of log10 [AG]X as a function of Tc - T is found to be roughly linear over the temperature range 0 to 50°C for this mixture, suggesting that the equation
iog10[^c7L = /iog 10 (r c -r) + J should be suitable. Suitable values for the constants /and /may be obtained by making the calculated value of [AG]X equal to the actual value at 0°C and at 50°C, whence / = 04462 and J = 0-0747.
198
PHASE EQUILIBRIUM IN MIXTURES
Values of [AG]X have been calculated using the equation log10[AG]x
= Ilog10(Tc
- T) + J
for each of the temperatures given on p. 196. "Residual values" obtained by subtracting these from the experimental values are shown plot ted as a function of temperature in Fig. 3*8. From the figure it is found that - ^ ( ^ G r c s l d u a l ) = 00055, whence —
(AG) = -
ca
*L + 0-0055 = - 0-0745 (1. atm)/(mole °K) at 30°C.
Hence, finally Y = 0-4709 - 00587 - 00745 = 0*3377 ± 0*0028 (1. atm)/(mole °K) and L = 592-2 ± 5 joules/mole.
EXAMPLE OF THE USE OF A RESIDUAL PLOT
Kozicki and Sage13 have determined the latent heat of vaporization (L) of n-pentane calorimetrically. Some of their experimental values are shown below. For manipulating their data they recommend the use of the "first order" equation L calc = (31700 - 82-1 0 1/2 where L is expressed in B.t.u./lb and t is the temperature in °F. Plot the resi dual latent heat (L(R) = L calc — L) as a function of temperature and hence obtain a value for the latent heat at 90°C. From an analysis of the sources of error, Kozicki and Sage consider that at temperatures less than 280°F their latent heats are subject to an uncertainty of ±0-2 per cent. Draw a vertical line corresponding to this uncertainty through each point on the residual plot. Some values for the latent heat of n-pentane calculated from PKTdata by Young14 are also shown below. Use the residual plot to decide whether or not these data agree with those of Kozicki and Sage within the experimental error claimed by these latter authors. Finally, calculate 18 14
/ . Chem. Eng. Data 5, 21, 331 (1960). Sci. Proc. Roy. Dublin Soc. 12, 374 (1909-10).
LATENT HEATS AND VAPOUR PRESSURES
199
aL — a° at 90°C. (Equation (3.12) may be used for this purpose. Write dL_ AT
dL c a l c dT
+
dL{R) dT
dL,calc is obtained from the "first order" equation and dL(R)/dT may dT be determined from the residual plot.) (a) Latent heats determined calorimetrically by Kozicki and Sage Temperature (°F) 1000 1300 1600 1900 2200 2500
Latent heat of vaporization B.t.u./lb 153-4 146-8 1400 132-8 124-5 115-3
(b) Latent heats calculated by Mills from vapour pressure and volumetric data of Young Temperature (°C) 30 40 50 60 70 80 90 100 110 120 130 140 150
Latent heat of vaporization cal/g 85-76 84-31 82-13 8007 77-77 75-33 72-73 69-94 67-31 64-48 60-85 56-58 52-39
COMPARISON OF EXPERIMENTAL AND CALCULATED LATENT HEATS In the case of pure substances, numerous experimental determinations of latent heat have been made, a typical apparatus being shown in Fig. 3 9. The calorimeter C is placed in an adiabatic jacket J, the space between the calorimeter and the jacket being evacuated to a pressure below 10 ~6 mm.
200
PHASE EQUILIBRIUM IN MIXTURES
The temperature of the evaporating liquid is determined using a resistance thermometer and the pressure is determined using a pressure balance. Heat is supplied to the calorimeter by the electrical heater D, the rate at which energy is supplied from this source being continuously noted. The quantity of liquid evaporated during a given period is determined by col lecting and weighing it. For further details of an apparatus of this type the reader is referred to Sage and Hough17 and also to Kozicki and Sage.13 It is normally found that the experimental and calculated latent heats agree within the expected experimental error. Indeed, the comparison of
KAww" Pressure balance Mercury separator
FIG. 3-9. Apparatus for determining latent heats at elevated pressures. The rate at which energy is supplied to the heater D is noted and the rate at which liquid condenses in the receivers B and B' is also noted. J is an evacuated jacket.
latent heat and vapour pressure data has been used as a means of calculating VG and hence the second virial coefficient B for vapours such as benzene vapour for which it is difficult to obtain such data directly.18 Thus the equation PV = RT + BP which represents the volumetric behaviour of a slightly imperfect gas may be combined with (3.4) to give B=
L dr
RT + Vl
(3.27)
"OSBORNE, STIMSON and FIOCK, / . Research Nat. Bur. Standards 5, 411 (1930). OSBORNE, STIMSON and GINNINGS, / . Research Nat. Bur. Standards 23,197 (1939).
16
17 18
SAGE and HOUGH, Anal. Chem. 22, 1304 (1950).
SCOTT, WADDINGTON, SMITH and HUFFMANN, / . Chem. Phys. 15, 565 (1947); TOMPA, J. Chem. Phys. 16, 292 (1948); ALLEN, EVERETT and PENNEY, Proc. Roy.
Soc. A 212, 149 (1952); PENNINGTON and KOBE, J. Am. Chem. Soc. 79, 300 (1957) (for acetone vapour); BARKER, BROWN and SMITH, Disc. Faraday Soc. No 15 (1953), 143 (for ethanol vapour).
LATENT HEATS AND VAPOUR PRESSURES
201
where L is an accurately determined experimental latent heat and VL is the volume of the liquid. The gradient dr/d/?°may be determined accurately, using the methods described in the previous section. EXTENSION OF VAPOUR PRESSURE DATA USING REFERENCE SUBSTANCE PLOTS If the vapour pressure of one substance has been accurately evaluated over a wide range of temperatures, these data may be used, with the aid of a suitable plot, to extend existing but more scanty data for other sub stances with similar properties. One possibility is to plot the temperaturat which the substance of interest has a given vapour pressure against the temperature at which the reference substance has the same vapour pressure. It is found that if the substance of interest is similar to the reference sub stance a roughly linear plot is obtained. If only scanty data are available for this substance, such a plot can assist in extending the data. This pro cedure was first proposed by Duhring. Rather straighter lines are usually obtained if the logarithm of the vapour pressure of the reference substance at a given temperature is plotted against the logarithm of the vapour pres sure of the substance of interest at the same temperature. This procedure was first proposed by Cox19 and has since been discussed by Othmer20 and by Calingaert and Davis.21 Using water as reference substance, it is found that a wide variety of substances, even dissimilar ones, give nearly straight lines when plotted by this method. It is found empirically that the Cox lines for groups of closely related compounds converge to single points which are characteristic of these groups. A single point of temperature was, for example, found for the paraffin hydrocarbons. For a member of such a group it is only necessary to know the vapour pressure of the member at one temperature and the convergence point in order to obtain an approxi mate vapour pressure curve for that substance over the entire temperature range. Probably a more accurate procedure for extending vapour pressure data is to plot the reciprocals of the absolute temperatures at which the substance of interest has given values of vapour pressure against the recipro cals of the absolute temperatures at which some similar reference substance has the same vapour pressure values. This type of plot has been used by Maxwell22 in his work on hydrocarbons. With one or two exceptions he found that the relationship was linear to within ±0-5°C over the entire range of the data. The reference compounds which he used for his work were themselves hydrocarbons. The most accurate procedure of all is probably to plot reciprocal absolute temperatures at the same reduced vapour pressures. 19
Ind. Eng. Chem. 15, 592 (1923). Ind. Eng. Chem. 32, 841 (1940). 21 Ind. Eng. Chem. 14, 1287 (1925). 22 Data Book on Hydrocarbons, Van Nostrand, 1950.
20
202
PHASE EQUILIBRIUM IN MIXTURES WATSON'S EMPIRICAL EQUATION FOR PREDICTING LATENT HEATS23
It was observed empirically on p. 191 that the "first order" equation log10 L = E log10 (Tc — T) + F provides a fairly accurate representation of latent heat data. It follows that L = B{\ - Tr)E where B and E are constants for a given substance and Tr is the reduced temperature. It follows that
Lx L2
Jl-Trl\E ( 1 - Tr2 j
where Lx and L2 are the latent heats of the given substance at the reduced temperatures Trl and Tr2- Watson23 has pointed out that E has much the same value for all substances, both polar and non-polar, which have been studied, being equal to 0*38. According to Watson's empirical equation therefore Li = (1 - Trl I0'38
L>
I1- TJ
In order to use the equation the critical temperature of the substance is required together with the latent heat at one temperature. The equation is simple to use and comparatively accurate. Errors do not normally exceed 5 per cent over the temperature range extending from the normal boiling point to a reduced temperature 0-85 and the equation remains reasonably accurate to within 10°C of the critical temperature.24
EFFECTS OF VAPOUR PHASE ASSOCIATION
(see also footnote 2)
Vapour phase association produces anomalously large values for Cp. The latent heats of associating substances may, over limited temperature ranges, rise with increase in temperature (the latent heats of normal sub stances fall with increase in temperature principally because Cp > Cp, cf. equation 3.19). The vapour pressure curves for such substances also show slightly anomalous behaviour. These effects have been considered quanti tatively by Armitage Gray and Wright25 for the substances formic acid, acetic acid and nitrogen dioxide (which dimerize to a certain extent in the vapour phase) and hydrogen fluoride (which polymerizes). 23 24 25
Watson, Ind. Eng. Chem. 35, 398 (1943). Reid and Sherwood, Properties of Gases and Liquids, p. 97, McGraw-Hill, 1958. Trans. Faraday Soc, 58,1746 (1962). / . Chem. Soc, March 1963, 1796, 1807.
LATENT HEATS AND VAPOUR PRESSURES Latent Heats. Definitions of latent heats for pure substances and mixtures, the Clausius-Clapeyron equation, analogues of the Clausius-Clapeyron equation applicable to mixtures, non-equilibrium latent heats, saturation specific heats. Vapour pressure equations for pure substances Wrede's equation, the Antoine equation, the Rankine-Kirchoff equation, Frost's equation. Temperature dependence of the dew- and bubble-point pressures of mixtures Accurate representation of vapour pressure and other phase boundary data using residual plots Use of residual plots for calculating latent heats from vapour pressure data Comparison of experimental and calculated latent heats Determination of second virial coefficients for vapours from latent heat and vapour pressure data. Extension of vapour pressure data
LATENT HEATS
The molar latent heat of a pure substance is defined as the heat required to convert one mole of that substance from the liquid to the vapour state under equilibrium conditions at constant temperature and hence constant pressure. Thus
S is the entropy of a closed system containing nG moles of gas phase. In the case of a mixture, the pressure and composition may change con tinuously during an isothermal evaporation and likewise the temperature and composition may change continuously during an isobaric evaporation process. In this case the heat required per mole of liquid converted may change continuously during the evaporation process. Suppose that one mole of a liquid mixture of specified composition is enclosed in a container and brought to the bubble point. The isothermal integral latent heat LT is defined as the heat required to convert the liquid completely to the vapour phase at constant temperature. Similarly the isobaric integral latent heat LF is the heat required to convert the liquid to the vapour phase at constant pressure. The differential molar latent heats at constant temperature and 175 13 PE (AP—250)
176
PHASE EQUILIBRIUM IN MIXTURES
constant pressure are defined by the equations
and
dS 1 dS \ / = T-= T- -—c\ T dnG\T dMGjTX
(3.1a) '
K
respectively. A/ G is the fraction of the moles of mixture which are present in the vapour phase and 5 is the molar entropy of the system. The isothermal integral and isobaric integral molar latent heats may be obtained from the differential latent heats using the equations I
LT=[lTdMG]TX
(3.2a)
LP = $lPdMG]PX
(3.2b)
and 6
lT and lP may be calculated from phase boundary data using equations (9.44) and (9.53) respectively. These equations are somewhat complex to use and detailed consideration of them will be deferred until Chapter 9. The isothermal integral latent heats of a series of mixtures of propylene and carbon dioxide are shown in Fig. 3*1.
THE CLAUSIUS-CLAPEYRON EQUATION FOR THE LATENT HEAT L OF A PURE SUBSTANCE
Consider a system containing one component only and consisting of a vapour phase in equilibrium with a liquid phase at pressure p° and tem perature T. For equilibrium, the chemical potential of this component, and hence the molar Gibbs function, must be the same in the two phases. The molar Gibbs function of any phase in a one-component system is a function only of the pressure and temperature, so for any infinitesimal change in the state of the phase dG
L-£L
dp = 4?"l ^rl dr dP )T + "T dTjp
= VdP-SdT
(3.3)
LATENT HEATS AND VAPOUR PRESSURES
177
Suppose that the temperature of the two-phase system changes from T to T + dT and the vapour pressure changes from p° to p° + dp°. The cor responding changes in the molar Gibbs function of the vapour and liquid phases are dGG and dGL and, since G° = GL before and after the change, dGG = dGL
Fio. 3*1. The isothermal integral latent heats of mixtures of carbon dioxide and propylene.
Combining this result with equation (3.3) Vcdp° - SGdT = FLd/?° - 5 L d T or L = T(SG - SL)=T(VG
- VL)^-
(3.4)
This is the well-known Clausius-Clapeyron equation. Provided care is taken in evaluating the slope dp°ldT of the vapour pressure line, this equa tion may be used to calculate latent heats with a precision equal to that attained by calorimetric methods. The determination of this slope is dis cussed further on pp. 189 — 95.
13*
178
PHASE EQUILIBRIUM IN MIXTURES
EQUATION FOR THE ISOTHERMAL INTEGRAL LATENT HEAT OF A MIXTURE
The isothermal integral latent heat may be calculated using the equation1 dP T
D.P.
dT)x
-v
B.P. ] B.P. dP +
dr
W([AG]X)X
(3.5)
i ^O^A, j
N.
N. S
If FIG. 3*2. Illustrating the derivation of equation (3.5) for the isothermal integral latent heat of a mixture.
[AG]X is the isothermal Gibbs function of condensation and is defined by the equation pu.r.
[AG}X=PS
VdP]TX
(3.6)
pD.P.
For a pure substance, P D P - = P a P - hence [AG]X = 0. F D R is the molar volume of the mixture at the dew point at the temperature T. VBF' is the molar volume at the bubble point. dP/dTf**' and dP/dT)BV- are respec tively the rate of change of the dew-point pressure and of the bubble-point pressure with temperature for a mixture of overall composition x. In the case of a pure substance, [AG]X = 0, dP/dT)^ = dP/d7)*p* = dp0/dT, and equation (3.5) reduces to the simple Clausius-Clapeyron equation L / r = (K DP - - VBJ,)dp°/dT
= {VG - VL)
dT
The validity of equation (3.5) may be demonstrated graphically as in Fig. 3*2. The full lines in this figure show the variation with pressure of the volume 1
HASELDEN, HOLLAND, KING and STRICKLAND-CONSTABLE, Proc. Roy, Soc. A 240,
13 (1957).
179
LATENT HEATS AND VAPOUR PRESSURES
of an arbitrary mixture in the two-phase region at some temperature T; the dotted lines show the state of affairs at a slightly higher temperature T + dr. The increase in temperature from T to T + dT is accompanied by changes not only in the volume at any given pressure but also in the values of the terminal pressures P DP - and P B P \ It is desired to determine the
FIG. 3*3. Illustrating the determination of the gradients required when using equation (3.5) to evaluate the isothermal latent heat of a mixture, (a) di7dr) DP - and djP/dr)BP- are obtained from thePTdiagram for the mixture in question, (b) The overall molar volume for the system is plotted as a funcpB.P.
tion of pressure at each of a series of temperatures and [A G]x = J VdP] is pD.P
obtained at each temperature by graphical integration, (c) d[A G]x/dT obtained from a plot of [AG]X as a function of temperature.
B.P.
difference d( J VdP) between the Gibbs function of condensation at the D.P.
higher temperature (which is the area enclosed between the dotted lines) and the Gibbs function of condensation at the lower temperature (which is the area enclosed between the full lines in Fig. 3*2). It is seen from the figure that the difference between these areas is equal to Ax+ A2- Az FB.P.
jpB.P.
+
|BJ' d K | D.P. dT
dp
j dT
i.e. to _
F D.P. dj pD.l
180
PHASE EQUILIBRIUM IN MIXTURES
where dP BP * and dPD,p* are respectively the increase in the pressure at the bubble point and the increase in pressure at the dew point when the temperature is increased from T to T + dT. Dividing by AT, it follows that B.P
dP)BP'
d/MDP-
D.P.
a i
Q1
IX
JX
BP
-
D.P.
dP. p
Since j 1
B.P.
Ay\
D.P.
QI
)P
1
B.P.
-i
\X
D.P.
ATX
the equation
follows at once. Provided that accurate volumetric data for the mixture are available within the two-phase region, equation (3.5) probably provides the most accurate method of calculating integral isothermal latent heats. The gradientsdP/dr) DR and dP/dr) B ' P, may be obtained from the PTdiagram for the mixture in question (Fig. 3*3a). The quantity [^G]^ may be obtained, at each of a set of temperatures, from equation (3.1), using a graphical technique to obtain the integral in each case (Fig. 3'3b). For this purpose, values of the overall volume of the mixture are required as a function of pressure at each temperature. Such data could in principle be obtained by enclosing a sample of the mixture in question in the apparatus shown in Fig. 2*1, and directly noting the total volume as a function of pressure. In order to evaluate the gradients dPDF/dT)x, dPBF/dT)x and d[AG]x/dT accurately, it is desirable to use residual plots. A worked example of such a calculation is given on p. 195, and the relative magnitudes of the terms
dT)x
dT)x
dT
for this case are shown. OTHER ANALOGUES OF THE CLAUSIUS-CLAPEYRON EQUATION APPLICABLE TO MIXTURES
A variety of equations may be used to calculate the latent heats of mix tures and all of these could be described as analogues of the ClausiusClapeyron equation. Equations (9.44) and (9.53) have already been men tioned. In addition, equations (9.30) and (9.41) have the same form as the Clausius-Clapeyron equation and may be used to calculate liquid phase
LATENT HEATS AND VAPOUR PRESSURES
181
enthalpies from those of the equilibrium vapour phase. Equation (3.6) differs from the above in that the data required when using it may be derived entirely from volumetric observations on the mixture for which the latent beat is required. No data for neighbouring mixtures are needed and a knowledge of the composition of the equilibrium gas and liquid phases is also unnecessary. SPECIFIC LATENT HEATS
The equations so far considered are for the molar latent heat. Analogous equations exist for the latent heat per unit weight of distillate. These may be obtained by substituting V(w) for V, x(w) for x, in the above equations. Thus equation (3.5) for the latent heat of a pure substance becomes L(w) = r [ r G ( w ) - K L ( v v ) ) - ^ ) and so on. (L(w) is the heat required to convert unit mass of substance from the vapour to the liquid state. VG(w) is the volume occupied by unit mass of vapour and VL(w) the volume occupied by unit mass of liquid at the temperature T and the vapour pressure p°. NON-EQUILIBRIUM LATENT HEATS
The latent heats considered above all refer to a phase change carried out in such a way that only infinitesimal displacements from equilibrium con ditions are allowed. Such conditions are never exactly satisfied. In practice, such deviations do not have an appreciable effect on the latent heat, except in the case of substances such as hydrogen fluoride which polymerize strongly in the gas phase, the degree of polymerization varying rapidly with the pressure.2 The heat of polymerization of hydrogen fluoride is large and it has been found that the non-equilibrium latent heat of vaporization at any given temperature does depend noticeably on the pressure at which evaporation is carried out. SATURATION SPECIFIC HEATS
The saturation specific heat a° for a vapour is the heat required to raise the temperature of unit quantity of vapour by unit amount, the pressure being adjusted so as to maintain the vapour at its dew point. Thus
2
SIMONS and HILDEBRAND, / . Am, Chem. Soc. 46, 2183 (1924).
182
PHASE EQUILIBRIUM IN MIXTURES
Similarly the saturation specific heat GL for a liquid is the heat required to raise the temperature of unit quantity of liquid by unit amount, the pressure being adjusted to maintain the liquid at its bubble point. Thus dS \BR
T
Since
(3 8)
•*- Hrl dS ;D-P- _ j)S_\ drj,
" dT)PX
+
-
J5S \
dP \DP»
dP)„
AT)X
'
it follows on combining (3.7) with (1.35) that 8VG\
dP lDJP-
c® is the constant pressure specific heat of the gas phase and the gradient dVG/dT)PX is the rate of change of volume with temperature at constant pressure in the gas phase, the gradient being taken at the dew point. Similarly, T JP dT) •"-*- r Tr),dfJ
<3 10>
-
Now
Lr=r(SDP-SBP) (3.11) D,P where LT is the isothermal latent heat, S - is the entropy of the system at the dew point and SBP* is the entropy of the system of the same com position at the same temperature, but at the bubble-point pressure. Dividing (3.11) by T, differentiating with respect to temperature and combining with (3.7) and (3.8),
Equations (3.7) to (3.12) are equally applicable to a pure substance or a mixture. VAPOUR PRESSURE EQUATIONS FOR PURE SUBSTANCES
The oldest and simplest equation for representing the vapour pressure of a pure substance as a function of temperature is due to Wrede.8 According to this equation, \og10p°=a-blT (3.13) *Ann.Phys. 53, 225(1841).
LATENT HEATS AND VAPOUR PRESSURES
183
Figure 3*5a on p. 192 shows the logarithm of the vapour pressure of propylene plotted as a function of the reciprocal of the absolute temperature from the normal boiling point to the critical point. It is found that throughout this extensive range the Wrede relationship represents the data to within ± 3 per cent. Although not rigidly accurate, the Wrede equation is, because of its simplicity, very useful as a "first order" equation and it is suitable for use in conjunction with a residual plot (see p. 189). Numerous other equations are in use for representing vapour pressure data for pure substances. The following may be mentioned. l°gio/>0== A -
B
KC + 0 (Antoine equation)
(3.14)
'ogiop°= c - d/T + elog T (Rankine-Kirchhoff equation) (3.15) log10 p° = f-
9IT - h log T + jP/T2 (Frost's equation)
(3.16)
The extent to which these and other equations in fact represent the experi mental data has been discussed by Waring.4 The Antoine and RankineKirchhoff equations are successful over a rather larger range of conditions than the Wrede equation but a four-parameter equation such as (3.16) is required to fit vapour pressure data satisfactorily from the triple point up to the critical point. The Wrede equation may be derived by integrating the Gausius-Clapeyron equation L = T(VG - VL) d(p°)/6T (3.4) if it is assumed that the latent heat L is constant, that the volume VG of the gas phase is given by the perfect gas law and that the volume F L of the liquid phase is negligible compared with VG. Putting VG = RT/p° =^ VG — FL, equation (3.4) becomes LdT _ Rdp°
~W
p°
Integrating for constant L, ]np°= -L/RT+
C
(3.17)
This is Wrede's equation with a = C/2-303 and b = L/2-303 R. All three of the assumptions upon which the above derivation is based break down completely as the critical point is approached. The approximate validity of the equation in the critical region must therefore be regarded as for* Ind. Eng. Chem. 46, 762 (1954).
184
PHASE EQUILIBRIUM IN MIXTURES
tuitous. In this connection, it is of interest to note that the ClausiusClapeyron equation may be rewritten in the form AT where Z G and Z L are the compressibility factors of the gas and liquid phases. Hence
d/>° p°
=
L
G
L
R{Z - Z )
G L J_ = _ (// -// )
T
1
1
(3.18)
"^(z^^z )" y
Provided the ratio L/(ZG — ZL) is independent of temperature, equation (3.18) may be integrated to give the Wrede equation, with b = L/2-303 R(ZG - ZL). The fact that Wrede's equation remains approxi mately true even up to the critical temperature indicates that the ratio L/(ZG — ZL) is approximately constant for a given substance. The Wrede equation represents vapour pressure data for solids quite adequately. Table 3.1 shows the constants a and b for a few substances in the solid state. The pressure range over which the equation is applicable and the precision with which the equation represents the experimental vapour TABLE 3.1.
CONSTANTS IN THE W R E D E EQUATION
(p° in m m of mercury, T in °K) Range (mm)
Substance Phenol tf-Cresol p-Cresol Acetylene Ammonia a
11-5638 12-7778 120298 8-8400 100059
3586-36 3970-17 3861-98 1127-1 1630-7
Precision (mm)
0-03-1-04 0-02-0-40 001-0-29 up to 962 7-45
+ 0-03 a ± 001a ± 0-009° _b
± 2 percent
BIDDISCOMBE and MARTIN, Trans. Faraday Soc. 54, 1316 (1958).
* BURRELL and ROBERTSON, / . Am. Chem. Soc. 37, 2188 (1915). e GIAUQUE and OVERSTREET, / . Am. Chem. Soc. 59, 254 (1937).
pressures are also shown. The Antoine equation* (3.14) is frequently used to represent vapour pressure data at low reduced pressures (up to say P r = 0-1). In this range it is rather more accurate than the simple Wrede equation. The constants may be derived by a method described by Willingham et al.5 The constants A, B and C are shown for various liquids in Table 3.2. * ANTOINE, Compt. rend. 107, 681 (1945). WILLINOHAM et al., J. Research Nat. Bur. Standards 35, 219 (1945).
6
185
LATENT HEATS AND VAPOUR PRESSURES TABLE 3.2.
CONSTANTS IN THE ANTOINE EQUATION
(p° in mm of mercury, t in °C) Substance Ethanol n-Propanol Triethylamine Phenol 0-Cresol /7-Cresol n-Pentane n-Hexane n-Decane Benzene Carbon tetrachloride
A 8-11219 7-9967 6-9952 7-13457 7-07055 7-11767 6-87372 6-87776 6-95367 1 6-89324 6-85906
Range (mm)
B
C
159219 1584-41 1300-30 1516-072 1542-299 1566029 1075-816 1171-530 1501-268 1203-835
226062 212-47 226-50 174-569 177-110 167-680 233-359 224-366 194-480 219-924
100-770 150-790 56-781 352-780 88-780 57-780 57-780
1196-60
224-138
90-770
90-770
Precision ± ± ± ± ± ± ± ± ± ±
0-16 mm' 0-03°C* 0-03°C* 0-007°C*«c 0-006°C*-c 0006°C**C 0-23 mm" 0-41 mmrf 0-25 mm* 0-24 mm
+ 0-21 mm
* Worst values.
a
BARKER, BROWN and SMITH, Disc. Faraday Soc. No 15, 143 (1953).
*COPP and FINDLAY, Trans. Faraday Soc. 56, 13 (1960). C BIDDISCOMBE and MARTIN, Trans. Faraday Soc. 54, 1316 (1958). d Ref. 5.
The Rankine-Kirchhoff equation (3*15) also represents vapour pressures of pure substances rather more accurately than Wrede's equation. Equation (3.15) may be deduced from the Clausius-Clapeyron equation, if a linear variation of the latent heat with temperature is assumed. The assumptions of an ideal gas phase and a liquid phase of negligible volume are also made as in the derivation of equation (3.17). It is convenient to write the latent heat in the form L = T(SG - SL) = (HG - HL) where HG is the molar enthalpy of the gas phase and / / L is the molar enthalpy of the equilibrium liquid. Differentiating the above expression for L with respect to tem perature, dL dT
d// G dT
dHL dT
It should be remembered that the gradients di/°/drand dHL/dT are not taken at constant pressure but along the saturation line. Thus dHG/dT is the rate at which the enthalpy of the saturated vapour phase changes with temperature. Now di/ G =
dH
err,
dinG
(
186
PHASE EQUILIBRIUM IN MIXTURES
and
dH\L
dHL
7 f I,
Hence dL ~dT
C
~ P-
_ dH iL dr + dH\c
C
P + \jp)T
8H\L]
-
TpjJ'dT
(3.19)
Except in the close vicinity of the critical point, the term(dHL/dP)T may be neglected and, if the gas phase is perfect, (dH/dP)T = 0. Under these conditions therefore, equation (3.19) reduces to dL 7 T
B
<119a)
^
where A cp is the difference between the constant pressure specific heats of the vapour and liquid phases. If it is assumed that A cp is independent of temperature, (3.19a) may be integrated to give L = L° + A cpT
(3.20)
The constant L° may be regarded as being the latent heat which the liquid would have if it could be subcooled to the absolute zero of temperature. Combining equations (3.20) and (3.4) and setting (V°-VL) = (RT/P°\ the equation dp° _ LpdT AcpdT p° "" RT2 + RT is obtained. Hence, on integrating,
y
RT
R
or
This expression for the vapour pressure is identical with equation (3.15), with d = (L°/2-303 R) and e=(AcpjR). At temperatures removed from the critical temperature, equation (3.15) is sufficiently accurate to summarize most available vapour pressure data within the experimental error. A table, giving the constants c, d and e for a number of liquids, has been given by
LATENT HHATS AND VAPOUR PRESSURES
187
Moelwyn-Hughes.6 It is found that the constant e is normally negative and that its absolute value increases with the molecular complexity. The constant d0 likewise increases with the size of the molecule, as is to be expected since it is directly proportional to L°, the latent heat extrapolated to a limiting low temperature. A further discussion of the dependence of the parameters c, d and e on the molecular structure is given by MoelwynHughes.7 Compared with the Antoine equation, the Kirchhoff equation suffers from the disadvantage that it is not possible to use it to calculate explicitly the boiling point temperature as a function of pressure. However, the Kirchhoff equation has the advantage that the constants in this equation have a physical interpretation. Neither the Kirchhoff nor the Antoine equation is suitable for use in the critical region. The Frost—Kalkwarf equation (3.16) may be used throughout the range from the triple point to the critical point. The equation was originally derived8 by integrating the Clausius-Clapeyron equation with the assump tions (a) that L is given by (3.20) with A c independent of temperature, (b) that V° — VL is correctly given by van der Waal's equation. However, the equation is probably best regarded as empirical. The first three terms are identical with those of the Kirchhoff equation; the last term is small at low reduced pressures but becomes large as the critical point is ap proached. In order to use the equation to calculate the vapour pressure it is necessary to employ an iterative method. One possible procedure is first to calculate the vapour pressure using the first three terms only; the value Px thus obtained may then be substituted into the last term and a new value P2 is calculated from the equation log P*=f-glT-
h log T + JPJT2
The above process is repeated until the difference (Px — P2) is negligibly small. Perry and Thodos9 have evaluated the constants f9g, h,j for the normal alkanes from methane to n-dodecane and have represented these constants as functions of the number of carbon atoms in the alkane chain. They obtain an average deviation in the calculated pressure of ±0-27 per cent. The constants/, g, h9j for a few hydrocarbons are shown in Table 3.3. The values for the unsaturated hydrocarbons are taken from a paper by Smith and Thodos.10 These authors list vapour pressure constants for many other unsaturated hydrocarbons and suggest a method by which the values of the constants may be estimated from a knowledge of molecular structure and the normal boiling point. 6
Physical Chemistry, pp. 699, 701, 2nd edition, Pergamon Press, 1965. Ref. 6, p. 696. 8 FROST and KALKWARF, / . Chem. Phys. 21, 264 (1953). 9 Ind. Eng. Chem. 44, 1649 (1952). 10 A. I. Ch. E. Journal 6, 569 (1960). 7
188
PHASE EQUILIBRIUM IN MIXTURES TABLE 3.3.
CONSTANTS IN THE FROST EQUATION
(p° in mm of mercury, T in °K) Substance Methane Ethane Propane n-Heptane n-Octane Ethylene Propylene 1-Heptene 1-Octene Acetylene Propyne
/
9
14-85689 16-51824 18-17125 28-10450 31-52935 1413307 1913914 25-90236 27-94965 18-37039 22-94034
592-41 1061-2 1397-4 2813-7 3215-5 997-69 1409-59 2659-36 2989-80 1188-90 1759-46
// 3-26150 3-48408 3-91480 6-87452 7-9220 3-75709 4-26637 6-15555 6-74267 403099 5-43367
J 0-23785 0-45265 0-70362 2-8875 3-8008 0-38247 0-70426 2-47607 303649 0-37407 0-43472
Example Use the Antoine equation to calculate the normal boiling point of phenol and also the latent heat of phenol at its normal boiling point. (See Table 3.2 and mention any assumptions which you make in your calculation.) (Answer: L = 16410 cal/mole) TEMPERATURE DEPENDENCE OF THE DEW- AND BUBBLE-POINT PRESSURES OF MIXTURES
Working over a sufficiently limited pressure range, it is possible to repre sent the dew- and bubble-point pressures and also the relative volatility for a mixture of fixed overall composition by the Wrede-type equations11 \ogiQP™=A-BIT]x log 10 / ,ap - = C-D/T]X log10a=E + F/T]x
(3.21) (3.22) (3.23)
The parameters A, B, C, D, may be plotted as functions of composition. This provides a convenient method of smoothing dew- and bubble-point data and also of interpolating the dew- and bubble-point pressures at com positions other than those for which they were determined experimentally. Unfortunately, as seen in the previous section, these equations are only valid over a rather limited range of conditions. For this reason, it is desirable to use the above equations in conjunction with residual plots as described in the next section. 11 DIN, Trans. Faraday So:. 56, 668 (1960). (System N 2 - 0 2 over pressure range 1-10 atm.)
LATENT HEATS AND VAPOUR PRESSURES
189
ACCURATE REPRESENTATION OF VAPOUR PRESSURE AND OTHER PHASE BOUNDARY DATA USING RESIDUAL PLOTS
If the primary vapour-liquid equilibrium data are to be used for sub sequent calculations or interpolations or even if they are to be checked for errors, it is necessary to find some means, analytical or graphical, of repre senting the data to the full degree of accuracy with which they were obtained.
/ /
(a)
i
X) 55
X
FIG. 3*4. Illustrating the construction of a residual plot. The purely analytical method is tedious, while the use of a direct graphical plot is frequently unsatisfactory since the data are usually obtained with an accuracy greater than that with which they can be represented on a piece of graph paper of reasonable size. Furthermore, even if a sufficiently large piece of graph paper is used, the spacing of the points may be large com pared with the curvature of the curves on which they lie, so that it is difficult to draw a best curve through the points or to see whether a point is "off line". A good example of a direct plot is given by Fig. 2*13, which shows the dew-point and bubble-point lines for mixtures of the system C 3 H 6 -C0 2 as a function of temperature. The experimental accuracy of the pressure measurements is claimed to be about ±0-01 atmosphere, while that of the temperature measurements is given as ±0-05°C. It is impossible, however, using the direct plot, to tell whether or not a given point actually lies on the curve to within these limits. The most satisfactory way of manipulating such data is to use the method of residual plots, which may be regarded as
190
PHASE EQUILIBRIUM IN MIXTURES
a combination of the analytical and graphical methods. As a first stage in using this method, an equation is found which approximately represents the experimental data (say of y as a function of x). It is convenient to call this equation the "first order equation". The small difference between the experimental and "calculated" values of the variable y is called the "residual" value of y and is plotted graphically as a function of x. This graph is called the residual plot. Examples of such plots are shown in Figs. 3*4, 3*5, 3*7 and 3*8. If a satisfactory form for the first order equation has been found, there should be no sharp points of inflexion and it should be possible to observe the experimental scatter on the residual plot. Since by definition, .Vres'dual
(x) (3.24) it is possible to interpolate a value for y at any given value for x. yca]c (x) is obtained from the first order equation and JVcsiduaiC*) *s r ^ a d from the residual plot. Gradients may similarly be determined more accurately than from the direct plot of y as a function of x. For this purpose, the equation
f -IfI +lf) ax
\ ax jcalc
\ ax jreSjdUai
<3-25)
is used. (dyldx)ca{c is obtained by differentiating the first order equation. (dj>/dx)reSidual may be determined from the residual plot. The error in this determination of (d>>/d;t)residual is unlikely to be less than ±5 per cent. However, since the magnitude of j>reSiduai s n o u ld normally be less than 1/10 that of y9 the error in the determination of dy/dx may be less than 0-5 per cent. The accuracy with which dy/dx may be determined by this method in any particular case depends on the accuracy of the data and on the extent to which these data are represented by the first order equation. When calculating (d}VdA')residuai, a tangent must be drawn to the residual curve. To facilitate this, a front-silvered (and preferably half-silvered) mirror and set-square may sometimes be required. The mirror is placed with its reflecting surface vertical to the plane of the graph paper and this surface is made to cut the curve at the point A at which the gradient is required. The mirror is now rotated about a vertical axis through A until no discontinuity can be seen between the gradient of the curve and its reflection at the point A. When this has been achieved, the actual curve and its reflection in the mirror should appear to lie on one continuous curve. The base of the mirror will then be perpendicular to a tangent drawn to the curve at the point A. The desired gradient may be determined using a transparent straight edge permanently attached to the back surface of the mirror. The mirror may be prepared from a flat glass plate, (a 2*/4 in. x 3V4 in. photographic plate with the emulsion removed is quite suitable) using the method of Twyman.12 Any deposit on the back side of the mirror 12
Trans. Optical Soc. 24, 203 (1922-3).
191
LATENT HEATS AND VAPOUR PRESSURES TABLE 3.4.
ANALYTICAL EQUATIONS WHICH APPROXIMATELY REPRESENT PHASE
EQUILIBRIUM DATA AND WHICH ARE SUITABLE FOR USE IN CONJUNCTION WITH A RESIDUAL PLOT
Variables
Quantities to be maintained constant
Equation
Dew-point, pressure and temperature
Composition
io gl0 P D - p - = . 4 - 2 ? / r A
Bubble-point pressure and temperature
Composition
Isothermal latent heat and temperature of vaporiza tion
log10i>BP-
Composition
Isothermal Gibbs function of condensation and tem perature
log 10 £r = £log 1 0 (Tc-T)+
Composition
Bubble-point pressure and composition
log 1 0 JG = / l o g
Temperature
=C-DjT]x
1 0
F
(r-D+/
i
should be removed. A beading placed along three sides of the mirror will protect the silvered surface. It should be noted however that in most cases the greatest contribution to the uncertainty in (dj/dx)ieS{dual will arise from the error involved in drawing the residual curve through the experimental points rather than from any error in determining the gradient of the curve as drawn. In these cases there is little point in using the mirror method to determine the gradient. Where possible, it is convenient to write the first order equation in the form
to) = ™/2(*) + c
(3.26)
where y is one of the experimental variables (say pressure or volume) and x is the other variable (say composition or temperature)./! and/ 2 are simple analytical functions such as log10 or reciprocal. The form of the functions / i and/ 2 may already be well known, otherwise intelligent guess-work must be used. Table 3.4 shows suitable forms for a number of cases. In order to see whether the guess for the form of/x and/ 2 is a good one, the experi mental values of x and y are used to calculate corresponding values of fx(x) and f2(y), and/i(x:) is then plotted against f2{y). If the first order equa tion (3.26) is a satisfactory one, this plot should be approximately linear, and should not show any sharp points of inflexion. If such a plot has been obtained, a straight line is drawn through the points, the gradient and intercept of this line providing the quantities m and c respectively in 14 AP-250
192
PHASE EQUILIBRIUM IN MIXTURES
(a)
2
n
1
3
r
Pressure, atm 10
15
20
30
40
-$>C-
\
£
X
B -04
./
\
-60
-40
-20
0 20 Temperature,
40 °C
(b)
t
r
60
80
FIG. 3-5. The vapour pressure of propylene.
LATENT HEATS AND VAPOUR PRESSURES
193
the first order equation. This may now be written down explicitly. As an example of the use of the method of residual plots, further consideration will now be given to the dew- and bubble-point pressures for mixtures of carbon dioxide and propylene. It is known that the vapour pressure of a pure substance is represented quite closely as a function of temperature by the equation log10/>° =a-b/T (3.13) When considering dew- and bubble-point pressures of a mixture, it would therefore seem reasonable, as afirstguess, to try the same form of equation. Figure 3"5a shows the logarithm of the vapour pressure of pure propylene
I
i
i
i
30
31
3-2
3-3
i
i
3-4 3-5 t/T, °K-'
i
i
i
l
3-6
3-7
3-8
3-9
FIG. 3*6. The logarithms of the dew-point and bubble-point pressures of a 50 mole per cent mixture of carbon dioxide and propylene shown as functions of the reciprocal of the absolute temperature.
plotted as a function of the reciprocal of the absolute temperature, and Fig. 3*6 shows the logarithms of the dew point pressure and of the bubblepoint pressure of a mixture of carbon dioxide and propylene containing 49-7 mole per cent C0 2 plotted as functions of the reciprocal of the absolute temperature. It is seen that except in the immediate vicinity of the critical point a nearly straight line relationship is obtained in each case. A similar result is obtained for the remaining mixtures indicating that equations (3.13), (3.21) and (3.22) are satisfactory for use as "first order" equations over a wide range of conditions. Values for the constants in these equations may conveniently be evaluated from the experimental points at the ex tremes of the range considered. For example, it is known that the bubblepoint pressure for the mixture containing 49-7 mole per cent carbon dioxide is 14-81 atm at -15°C and 57-98 atm at 50°C So log10PBP- = 1-1706 for 14*
194
PHASE EQUILIBRIUM IN MIXTURES
l/r = (3-8737 x l O - y K - 1 and log 10 P BP ' = 1-7633 for 1/r = (3-0944 x x lO-yK- 1 . Hence suitable values to chose for D and C in equation (3.22) 1-7633-11706 = 0-7606x3-0944 and = 3-8737-3-0944 = 4-1169. In a similar way it is found that the equation log10 p° = 4-3380 -978-0/rshould be suitable for use as afirstorder equation when represent ing the vapour pressure of pure propylene and that the equation log10 PUFt = 4-7648 - 1014-5/r should be a satisfactory first order equation to use when representing the dew-point pressure of the above mixture as a function
__! -10
1 0
I 10
I 20 Temperature,
I 30
1 40
I 50
I 60
°C
FIG 3*7. Residual plots giving the dew-point and bubble-point pressures of a 50 mole per cent mixture of carbon dioxide and propylene as functions of temperature. These plots are to be used in conjunction with the equations log 10 P u? ' = 4-7648-1014-5/r and log10/>B*p- = 4-1169- 760-6/r where P is expressed in atm and Tin °K.
of temperature. Residual plots based on these equations are shown in Fig. 3-7. The experimental uncertainty in the pressure is estimated to be ±0-01 atm and this uncertainty is indicated by a vertical line through each point on the residual plots. No great care is required in the evaluation of the constants in thefirstorder equation, since it is sufficient that the equation should approximately represent the experimental data.
195
LATENT HEATS AND VAPOUR PRESSURES ILLUSTRATIONS OF THE USE OF RESIDUAL PLOTS IN LATENT HEAT CALCULATIONS
Illustration 1 Calculate the latent heat of pure propylene at 30°C, given that the volumes of the saturated gas and liquid phases at this temperature are respectively 1-5233 and 0-0858 l./mole. From equation (3.4), OLv) L=T(VG-V
L v
d
^
dT
Now dp°
d/>°calc
d/£esidual
dT
dT
dT
where
dT
( /&lc ) I d l o g /?calc -[-%■][ \ T }{ d(l/T) 10
2
x 2-303
Using the equation logl0 p° = 4-3378 — 978-0/J and the corresponding residual plot, it follows that at 30°C
Jt
dr
A* 7 1 O ^
IV OOl
S\ XV
AK
A J U J —• v J I / 1
an
OT/-1
and that Residual
dT
= - 0 0 0 7 8 ± 0008 atm °K- 1
Hence ^30°C
= 1-4375 x 0-3092 = 134-75 ± 0*4 1., atm/mole = 779 ± 2 joules,toole Illustration 2
Calculate the isothermal latent heat of condensation of a 50 mole per cent mixture of propylene and carbon dioxide at 30°C, given that the molar volumes of the mixture at the dew-point and bubble-point pressures are 0-7185 and 0-0754 l./mole respectively. The isothermal Gibbs function of condensation for this mixture has been evaluated by graphical integration
196
PHASE EQUILIBRIUM IN MIXTURES
(see Fig. 3#3b). It is found to vary with temperature as follows
t°C 15 0 15 30 40 50
[AG]X (1. atm/mole) 8-174 7-197 6-291 5-287 4-458 3-377
The critical or "plait point" temperature for the mixture is known to be 60-4°C. Method From equation (3.5), T
dT)x
AT)X
The term I/D.P.
dr
U
lxJX
d/> fDPdT
may be written equal to i/D.P. I
dpD.P. calc
KXA
dT
,
dp®?UA residual
dT
where P^ic* *s gi v e n by the equation logio^aic* = 476478 - (1-01445 x 10 3 )/r and Pje'sTciuai *s shown plotted as a function of temperature in Fig. 3*7. dPcDaicp/dr = p ° £ d io g e pcDalcp/dr 1-01445 = 2-3026 (antilog (476478 - 101445/7)) x — - ^ ~ so at 30° dPcDalp/dr = 2-3026 x26-21 2 x 1-01445 x 10-881 = 0-6662 atm "K-1. From Fig. 37, dP%MUJdT = -0-0108 ± -001 atm °K-\ Hence FD-p-
j p
F BP - dP/dT)V- = 0-0754 W~r- + —
r
\
^ L
LATENT HEATS AND VAPOUR PRESSURES
197
where log10P^te = 4-11685 — 0-76058/r and P^duai i s shown as a func tion of temperature in Fig. 3*7. From this plot it is found that dPf^^JdT = 0-0054 atm 0 K~ 1 at 30°C Hence V**dPldT)y- = 0-0754(0-3357 x 2303 + 00054) = 00587 ± 00001 (1. atm)/(mole °K) In order to evaluate the quantity (dldT)([AG]x), it is desirable to obtain a suitable first order equation representing the variation of [AG]X with
-20
-10
10
20
Temperature,
30 °C
FIG. 3*8. The residual isothermal Gibbs function of condensation for a 50 mole per cent mixture of carbon dioxide and propylene. This graph is to be used in conjunction with the equations [AGh^AG^ and
+ A (7rcsldual
log10 AG^ = 0-4462 log10 (333-6-T) + 0*0747
temperature. A plot of log10 [AG]X as a function of Tc - T is found to be roughly linear over the temperature range 0 to 50°C for this mixture, suggesting that the equation
iog10[^c7L = /iog 10 (r c -r) + J should be suitable. Suitable values for the constants /and /may be obtained by making the calculated value of [AG]X equal to the actual value at 0°C and at 50°C, whence / = 04462 and J = 0-0747.
198
PHASE EQUILIBRIUM IN MIXTURES
Values of [AG]X have been calculated using the equation log10[AG]x
= Ilog10(Tc
- T) + J
for each of the temperatures given on p. 196. "Residual values" obtained by subtracting these from the experimental values are shown plot ted as a function of temperature in Fig. 3*8. From the figure it is found that - ^ ( ^ G r c s l d u a l ) = 00055, whence —
(AG) = -
ca
*L + 0-0055 = - 0-0745 (1. atm)/(mole °K) at 30°C.
Hence, finally Y = 0-4709 - 00587 - 00745 = 0*3377 ± 0*0028 (1. atm)/(mole °K) and L = 592-2 ± 5 joules/mole.
EXAMPLE OF THE USE OF A RESIDUAL PLOT
Kozicki and Sage13 have determined the latent heat of vaporization (L) of n-pentane calorimetrically. Some of their experimental values are shown below. For manipulating their data they recommend the use of the "first order" equation L calc = (31700 - 82-1 0 1/2 where L is expressed in B.t.u./lb and t is the temperature in °F. Plot the resi dual latent heat (L(R) = L calc — L) as a function of temperature and hence obtain a value for the latent heat at 90°C. From an analysis of the sources of error, Kozicki and Sage consider that at temperatures less than 280°F their latent heats are subject to an uncertainty of ±0-2 per cent. Draw a vertical line corresponding to this uncertainty through each point on the residual plot. Some values for the latent heat of n-pentane calculated from PKTdata by Young14 are also shown below. Use the residual plot to decide whether or not these data agree with those of Kozicki and Sage within the experimental error claimed by these latter authors. Finally, calculate 18 14
/ . Chem. Eng. Data 5, 21, 331 (1960). Sci. Proc. Roy. Dublin Soc. 12, 374 (1909-10).
LATENT HEATS AND VAPOUR PRESSURES
199
aL — a° at 90°C. (Equation (3.12) may be used for this purpose. Write dL_ AT
dL c a l c dT
+
dL{R) dT
dL,calc is obtained from the "first order" equation and dL(R)/dT may dT be determined from the residual plot.) (a) Latent heats determined calorimetrically by Kozicki and Sage Temperature (°F) 1000 1300 1600 1900 2200 2500
Latent heat of vaporization B.t.u./lb 153-4 146-8 1400 132-8 124-5 115-3
(b) Latent heats calculated by Mills from vapour pressure and volumetric data of Young Temperature (°C) 30 40 50 60 70 80 90 100 110 120 130 140 150
Latent heat of vaporization cal/g 85-76 84-31 82-13 8007 77-77 75-33 72-73 69-94 67-31 64-48 60-85 56-58 52-39
COMPARISON OF EXPERIMENTAL AND CALCULATED LATENT HEATS In the case of pure substances, numerous experimental determinations of latent heat have been made, a typical apparatus being shown in Fig. 3 9. The calorimeter C is placed in an adiabatic jacket J, the space between the calorimeter and the jacket being evacuated to a pressure below 10 ~6 mm.
200
PHASE EQUILIBRIUM IN MIXTURES
The temperature of the evaporating liquid is determined using a resistance thermometer and the pressure is determined using a pressure balance. Heat is supplied to the calorimeter by the electrical heater D, the rate at which energy is supplied from this source being continuously noted. The quantity of liquid evaporated during a given period is determined by col lecting and weighing it. For further details of an apparatus of this type the reader is referred to Sage and Hough17 and also to Kozicki and Sage.13 It is normally found that the experimental and calculated latent heats agree within the expected experimental error. Indeed, the comparison of
KAww" Pressure balance Mercury separator
FIG. 3-9. Apparatus for determining latent heats at elevated pressures. The rate at which energy is supplied to the heater D is noted and the rate at which liquid condenses in the receivers B and B' is also noted. J is an evacuated jacket.
latent heat and vapour pressure data has been used as a means of calculating VG and hence the second virial coefficient B for vapours such as benzene vapour for which it is difficult to obtain such data directly.18 Thus the equation PV = RT + BP which represents the volumetric behaviour of a slightly imperfect gas may be combined with (3.4) to give B=
L dr
RT + Vl
(3.27)
"OSBORNE, STIMSON and FIOCK, / . Research Nat. Bur. Standards 5, 411 (1930). OSBORNE, STIMSON and GINNINGS, / . Research Nat. Bur. Standards 23,197 (1939).
16
17 18
SAGE and HOUGH, Anal. Chem. 22, 1304 (1950).
SCOTT, WADDINGTON, SMITH and HUFFMANN, / . Chem. Phys. 15, 565 (1947); TOMPA, J. Chem. Phys. 16, 292 (1948); ALLEN, EVERETT and PENNEY, Proc. Roy.
Soc. A 212, 149 (1952); PENNINGTON and KOBE, J. Am. Chem. Soc. 79, 300 (1957) (for acetone vapour); BARKER, BROWN and SMITH, Disc. Faraday Soc. No 15 (1953), 143 (for ethanol vapour).
LATENT HEATS AND VAPOUR PRESSURES
201
where L is an accurately determined experimental latent heat and VL is the volume of the liquid. The gradient dr/d/?°may be determined accurately, using the methods described in the previous section. EXTENSION OF VAPOUR PRESSURE DATA USING REFERENCE SUBSTANCE PLOTS If the vapour pressure of one substance has been accurately evaluated over a wide range of temperatures, these data may be used, with the aid of a suitable plot, to extend existing but more scanty data for other sub stances with similar properties. One possibility is to plot the temperaturat which the substance of interest has a given vapour pressure against the temperature at which the reference substance has the same vapour pressure. It is found that if the substance of interest is similar to the reference sub stance a roughly linear plot is obtained. If only scanty data are available for this substance, such a plot can assist in extending the data. This pro cedure was first proposed by Duhring. Rather straighter lines are usually obtained if the logarithm of the vapour pressure of the reference substance at a given temperature is plotted against the logarithm of the vapour pres sure of the substance of interest at the same temperature. This procedure was first proposed by Cox19 and has since been discussed by Othmer20 and by Calingaert and Davis.21 Using water as reference substance, it is found that a wide variety of substances, even dissimilar ones, give nearly straight lines when plotted by this method. It is found empirically that the Cox lines for groups of closely related compounds converge to single points which are characteristic of these groups. A single point of temperature was, for example, found for the paraffin hydrocarbons. For a member of such a group it is only necessary to know the vapour pressure of the member at one temperature and the convergence point in order to obtain an approxi mate vapour pressure curve for that substance over the entire temperature range. Probably a more accurate procedure for extending vapour pressure data is to plot the reciprocals of the absolute temperatures at which the substance of interest has given values of vapour pressure against the recipro cals of the absolute temperatures at which some similar reference substance has the same vapour pressure values. This type of plot has been used by Maxwell22 in his work on hydrocarbons. With one or two exceptions he found that the relationship was linear to within ±0-5°C over the entire range of the data. The reference compounds which he used for his work were themselves hydrocarbons. The most accurate procedure of all is probably to plot reciprocal absolute temperatures at the same reduced vapour pressures. 19
Ind. Eng. Chem. 15, 592 (1923). Ind. Eng. Chem. 32, 841 (1940). 21 Ind. Eng. Chem. 14, 1287 (1925). 22 Data Book on Hydrocarbons, Van Nostrand, 1950.
20
202
PHASE EQUILIBRIUM IN MIXTURES WATSON'S EMPIRICAL EQUATION FOR PREDICTING LATENT HEATS23
It was observed empirically on p. 191 that the "first order" equation log10 L = E log10 (Tc — T) + F provides a fairly accurate representation of latent heat data. It follows that L = B{\ - Tr)E where B and E are constants for a given substance and Tr is the reduced temperature. It follows that
Lx L2
Jl-Trl\E ( 1 - Tr2 j
where Lx and L2 are the latent heats of the given substance at the reduced temperatures Trl and Tr2- Watson23 has pointed out that E has much the same value for all substances, both polar and non-polar, which have been studied, being equal to 0*38. According to Watson's empirical equation therefore Li = (1 - Trl I0'38
L>
I1- TJ
In order to use the equation the critical temperature of the substance is required together with the latent heat at one temperature. The equation is simple to use and comparatively accurate. Errors do not normally exceed 5 per cent over the temperature range extending from the normal boiling point to a reduced temperature 0-85 and the equation remains reasonably accurate to within 10°C of the critical temperature.24
EFFECTS OF VAPOUR PHASE ASSOCIATION
(see also footnote 2)
Vapour phase association produces anomalously large values for Cp. The latent heats of associating substances may, over limited temperature ranges, rise with increase in temperature (the latent heats of normal sub stances fall with increase in temperature principally because Cp > Cp, cf. equation 3.19). The vapour pressure curves for such substances also show slightly anomalous behaviour. These effects have been considered quanti tatively by Armitage Gray and Wright25 for the substances formic acid, acetic acid and nitrogen dioxide (which dimerize to a certain extent in the vapour phase) and hydrogen fluoride (which polymerizes). 23 24 25
Watson, Ind. Eng. Chem. 35, 398 (1943). Reid and Sherwood, Properties of Gases and Liquids, p. 97, McGraw-Hill, 1958. Trans. Faraday Soc, 58,1746 (1962). / . Chem. Soc, March 1963, 1796, 1807.