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Heat-loss corrections for small isoperibol-calorimeter reaction vessels
J. Chem. Thermodynamics 1975,7,919-926
Heat-loss calorimeter
corrections reaction
for small isoperibolvessels a
L. D. HANSEN, T. E. JENSEN, S. MAYNE, R. M. IZATT, and J. J. CHRISTENSEN
D. J. EATOUGH,
Departments of Chemistry and Chemical Engineering, and Contribution No. 59 from the Center for Thermochemical Studies, Brigham Young University, Provo, Utah 84602, U.S.A. (Received 13 June 1974; in revisedform
16 October 1974)
A more exact method of calculating the heat loss from isoperibol-calorimeter reaction vessels is described. This method involves corrections for changes in the heat-leak constant and the external power input with changes in the calorimeter contents, and is particularly important for reaction vessels with volumes of less than 25 cm3 or with heat-leak constants greater than about 0.005 min-I. The new calculation has been tested by application to results collected with a titration calorimeter of capacity 3 cm3.
1. Introduction Titration calorimetry has proven to be an especially useful technique for the determination of AH where simultaneous reactions exist,(‘) for the “determination” of AG for reactions involving weakly interacting species, (2-4) for the identification of species present in complex reaction mixtures,(l, 5-7) and for analytical determinations.(5-7) By reducing the size of the reaction vessel, the technique can be extended to many systems where materials are rare, expensive, or available only in limited amounts.(8P 9, Reduction of the volume of solution at a given concentration means that less material need be used while the same accuracy of determination is maintained; or, reduction of volume for a given amount of solute means that more concentrated solutions can be prepared, resulting in larger temperature changes and increased accuracy. These effects arise because while the actual enthalpy change is proportional to the extent of reaction, the temperature change is inversely proportional to the volume. Although analytical thermometric titrations have been done in as little as 5 cm3 of solution for a number of years, (5-7) this paper reports the first complete study of the requirements for isoperibol-calorimetric titrations with volumes of less than 5 cm3. The heat leakage to the surroundings is proportionately much greater from small (c 25 cm3) than from large (> 25 cm3) reaction vessels and more exact equations than previously derivedC3’ are required to correct for heat loss during the experiment. This arises because the heat leak modulus is an inverse function of the volume of the reaction vessel since there is a practical limit to the reduction in size of wires, stirrer a Supported by Grants GM18816 and AM15616 from the U.S. Public Health Service, and by NSF Grant GP33536X. 61
920
ET AL,.
L. D. HANSEN
shaft, and reaction vessel walls which constitute the connections between the system and its surroundings. Heat-leak constants, IC,as measured for various sizes of Dewar vessels specially constructed to have small vaiues of rc and used in our laboratory in the past are plotted against the volume of the reaction vessel in figure 1. It can be 0.04*
0.03‘; .; 0.02\ Y
o.ol-\
.;
;
1
'1:
60
80
100
l -• 0
20
a 40
Y/cm3 FIGURE 1. Modulus of heat leak K for specially constructed glass Dewar reaction vessels as a function of Dewar vessel volume V.
seen that as the volume of the Dewar vessel decreases the value of rc becomes greater and the heat loss to the surroundings during a titration could be equal to or even exceed the isothermal enthalpy change resulting from the chemical reaction. The relative errors resulting from the assumptions previously used to correct for heat loss then become larger and a more exact description of the heat loss must be used. This paper presents the derivation of equations accurately describing heat exchange between an isoperibol-calorimeter reaction vessel and its surroundings and the testing of those equations with experimental results.
2. Experimental EQUIPMENT
The calorimeter used in this study was a Tronac head and a 3 cm3 glass Dewar reaction vessel. thermistor bridge was (114.5kO.5) mV K -l. The meter syringe driven by a stepping motor. The detail elsewhere.“’
Model 1000 equipped with a small The temperature sensitivity of the buret was a 1 cm3 Gihnont microcalorimeter has been described in
MATERIALS TRIS solutions
were prepared by mass from TRIS (Fisher, primary standard) dried at 383 K. HCl (DuPont, reagent) solutions were standardized both by pH titration against TRIS and by Mohr titration with AgNO, (Baker, reagent). NaOH (B & A, 50 per cent, carbonate free reagent) solutions were standardized against the HCI solutions by pH titration. All water was distilled, filtered through charcoal, treated with a mixed bed ion exchange column, and freshly boiled before use.
HEAT-LOSS
CORRECTIONS
FOR SMALL
VESSELS
921
CALIBRATION
The energy equivalent E of the calorimeter was determined as a function of heating rate, heating time, and volume of water. The value of E did not vary significantly with either heating rate or time, indicating that no significant amount of heat escaped through the electrical leads of the heater. A plot of E against volume of water in the Dewar vessel was linear giving further evidence that there is no significant systematic error in the measurement of E by electrical heating. The energy equivalent of the empty vessel, (0.181+ 0.009) Cal,, K-l, was obtained by subtracting the heat capacity of the water from the total energy equivalent measured at each volume and averaging the results.? MEASUREMENTS For both enthalpy-of-reaction and energy-equivalent determinations, readings were taken at either 4 or 10 s intervals, the choice of interval length depending on the total length of time the buret or heater was operating. The total time was either 6.7 or 16.5 min. From 60 to 99 readings were taken while the buret or heater was on and one-fourth of these readings were taken during both the lead and trial periods. For chemical runs, 0.25 cm3 of either 0.25 or 0.07 M HCl was titrated, respectively, into 2.69 cm3 of 0.01 or 0.003 M NaOH or TRIS solutions. The more concentrated solutions produced changes of about 270 mcalti, and the more dilute about 100 meal,, during the titrations, corresponding to temperature rises of about 0.08 and 0.03 K, respectively. Bridge unbalance potentials were measured with a digital voltmeter with a resolution of ) 1 pV.
3. Calculations ANALYSIS OF THERMOGRAMS Equations have been presented (3) for the analysis of results of calorimetric titrations which include methods for calculating: (1) the total enthalpy change produced in the reaction vessel; (2) the correction terms due to effects from non-chemical energy; (3) the effect resulting from the difference in temperature between the titrant and titrate; (4) the ‘enthalpy of dilution of the titrant and titrate; and (5) the enthalpy change from chemical reactions other than the one of interest. From these quantities the enthalpy change of the particular reactions of interest can be calculated. The equations in reference 3 are all applicable to small reaction vessels except for those used in calculating the correction terms due to effects from non-chemical energy (2 above). These effects include stirring of the solution, heat exchange between the reaction vessel and its surroundings, Joule heating by the thermistor, and evaporation of the liquid in the reaction vessel. An equation which is valid for correcting for heat losses from reaction vessels with large heat-leak constants can be derived as follows. The rates of temperature change in the Dewar during the lead, run, and trial periods (subscripts 1 and t refer to the midpoints in time for the lead and trial periods t Throughout
this paper calm = 4.184 J; M = mol dm-+.
922
L. D. HANSEN
ET
AL.
and p to any point in the run period) caused by heat exchange with the surroundings are given by:flO) (dT/dt), = P,/si+~i(Tj - T,); (1) (dT/dt), = Pp/ap+icp(Tj- TP); (2) (dT/dt), = P&+K& - TJ; (3) where P denotes the total external power inputs, E the energy equivalents, rc the cooling constants, Tj the temperature of the surroundings, and T,, TP, and Tt the temperatures in the three periods. Multiplying (l), (2), and (3) by the respective thermal equivalent in each region and using the convention that heat input to the calorimeter is given a negative sign gives for the rates of heat exchange qHL: = -P,-slrcl(Tj-Q; qHL,I = -(dT/dt)ie, (4) -(dT/dt),s, = -P,-&pl~p(Tj-TTp); qHL,p = (5) 4 HL,t = -(dT/dt),s, = -Pt-&Tj-ZJ. (6) At this point in previous derivationst3) it has been assumed that TX = T,, Ty = T,, lcl = rcP = rcc,,PI = P, = P,, and E,I$ = ERIC,, = E,IC,, where subscripts x and y refer to times when titrant first enters and when the buret is turned off, and using these assumptions (4), (5), and (6) have been combined to obtain an equation for qm, P.(3) These simplifying assumptions are sufficient for Dewar vessels larger than 25 cm3 and with values of K less than about 0.005 mine1 in which the correction for heat exchange with the surroundings is a small fraction of the total change measured. For the Dewar vessel used in this study (3 cm3 volume, rc = 0.046 min-I). The above assumptions resulted in about 10 per cent error, nine-tenths of which arose from ElSSUUllIlg
EIJC,
=
EplCp
=
E,IC,.
In order to solve equations (4), (5), and (6) for reaction-vessel volumes less than 25 cm3 it is necessary to know how rc varies with volume. The dependence of K on volume can be determined by making electrical calibrations at different volumes. Since the volume of the liquid in the Dewar does not change during a calibration run, PI = P,, Ed = E,, rci = K~,and equations (1) and (3) may be combined and solved for K to give K = ((dT/dt),-(dT/dt),}/(T,TJ. (7) A plot of K against volume of water in the 3 cm3 Dewar vessel used in this study is shown in figure 2 and may be approximated as a straight line up to 3 cm3 at which point the value of IC rises rapidly. (*) For simplicity it is better to work at volumes included in the linear region. We thus write equations (8) and (9) for a titration performed within this linear region: Kp = 7c,+au,; (8) 7c,= Jc*+cw,; (9) where 0 is the volume of titrant added and cxthe rate of change of K with ZJ. Combining equations (4), (5), and (6) with (8) and (9) gives: ~HL,I
(10)
=-~I-EIKI(T~-Q
4 HI,, p = - p, - E&G &IL,
t =
-
p,
-
@I
+ @u,)(Tj + Nvj
- T,), -
m
(11)
(12)
HEAT-LOSS CORRECTIONS 0.150,
I
I
I
0.125 -7.c: o.loo _ d&dY = 0.0099 mine1cme3 ,I E >
923
FOR SMALL VESSELS I P I _ ;’
*’
0.075.a’
\
0.050 o.0252.0
4-a
-M
9’
”
2.5
3.0
4.0
3.5I
V/cm3 FIGURE 2. Modulus of heat leak K for a Dewar vessel, volume 3 cm3, as a function of the volume V of water in the vessel.
which contain six quantities, P,, P,, Pt, IC,, Tj, and &u, p, the values of which are not available from the experimental results collected during a titration run. Values must be assigned to three of the quantities to allow calculation of qHL, p. Tj may be approximated by setting it equal to the bath temperature. This is an approximation because there are thermal connections to the liquid in the Dewar vessel which may not be at bath temperature, i.e. the stirrer shaft and the electrical leads. (Alternatively an absolute value of rcclcould be determined from the results of electrical-heating calibration runs; however, it was found that slightly more precise values of AH were obtained using the Tj approximation.) The value of P at a given total liquid volume in the Dewar vessel may be found by plotting qHL, , against (q - IQ. The intercept at (q - T) = 0 is -P (see equation 10). By repeating these calculations at several different volumes of liquid in the reaction vessel it is possible to determine how P varies with the liquid volume. For the calorimeter used in this study P was found to be a linear function of volume@) so we write : p, = p, + PQ’ (13) P, = P, + ~~*, (14) where p is the rate of change of P with the volume of titrant v, and where equations (13) and (14) give the change in external power input as the total volume of liquid in the Dewar changes. The expression for qHL,p in terms of experimentally measurable quantities is obtained by combining equations (IO), (1 l), and (12) with (13) and (14), defining 8, = (Tj - If’,), 8, = (Tj - T,), and 8, = (Tj - T,), and eliminating P, and rcr: qHL,
p =
@HL,
I -
&,
-
Epf=Jp +
ikpep
0,) -
&1 6hwt
-
+ fmkHL,
E,, and E, can be expressed as linear functions of substitution of EP= %fPp, Et = -%+I%
t -
Ed
qHL,
1 +b+
Et~~,4).
(15)
and ZI (see section 2), but the ‘36)
(17)
924
L. D. HANSEN
ET AL.
where y is the rate of change of E with o, into (15) need not be done since in actual application numerical values of ep and at are obtained and substituted. The total heat exchange Qmp between the reaction vessel and the surroundings from the beginning of the titration toppoint p can be calculated from the equation: tP
QHL,p"
s tx
(18)
qHL dt.
The procedures used for this integration have been described in detail.(3) All other calculations involved in the analysis of the thermograms were the same as previously described.(3) CALCULATION
OF AH
Values of AH for the protonation of TRIS and the ionization of water were obtained by a linear least-squares fit of the corrected enthalpy change Qc,, with the amount of titrant added at selected points in the thermogram.(3) The points selected were those in the middle two-thirds of the reaction region. These were selected in order to avoid the problem of reaction of the titrant with any carbonate or hydrogen carbonate ions present in the solutions. Corrections for the enthalpy of dilution of the titrant were made using data from reference 11. All calculations were done on an IBM 7030 computer and FORTRAN programs are available.
4. Results and discussion of
The final results, -(11.35+0.05) TRIS (18 runs) and (13.35F0.05)
TABLE
kcal, mol-l for the enthalpy of protonation kcal,, mol -I for the enthalpy of ionization of
1. The enthalpy of protonation of TRIS with HCl at 298.15 K as determined in a 3 cm3 isoperibol titration calorimeter (cal$,, = 4.184 J; M = mol dmP3) -AH/k&,, mol-l o(AH/k&, mol-I) 0.2581 M HCl into 0.00919 M TRIS 11.24 0.01 11.17 0.01 11.28 0.01 11.32 0.01 11.19 0.02 11.22 0.01 0.2581 M HCI into 0.00903 M TRIS 11.34 0.09 11.46 0.11 11.24 0.06 11.29 0.08
Average at I, = 0.009 M: 11.28 f 0.03 b
a
-AH/k&,, mol-1 o(AH/kcalth mol-I) 0.06727 M HCl into 0.00313 M TRIS 11.73 11.14 11.44 11.05 11.31 11.55 11.64 11.68 Average at I, = 0.003 M: 11.44 f 0.09 b
0.04 Grand average: 11.35 i 0.05 c
DStandard deviation within a run. b Standard deviation of the mean among runs in the set. c Standard deviation of the mean among all runs.
0.05 0.03 0.02 0.05 0.05 0.08 0.09 0.08 0.06
a
HEAT-LOSS
CORRECTIONS
FOR SMALL
925
VESSELS
Hz0 (14 runs), are in agreement with accepted values (i.e. - 11.32 and 13.35 kcal,, mol-l, respectively).(11,‘2) The deviations both within a run and among the runs indicated that the overall accuracy of the AH obtained in this study is about +O. 1 kcal,, mol-I. (See tables 1 and 2 for a list of the values of AH and the standard deviations obtained.) The agreement between the accepted values and the values obtained in this study show that we have successfully described the heat loss from the Dewar vessel in these TABLE
2. The enthalpy of ionization of water as determined in a 3 cm3 isoperibol titration calorimeter (Cal,, = 4.184 J; M = mol dmm3)
- AZZ/kcaltb moi - 1 o(AZZ/kcalu, mol -I) a 0.2581 M HCI into 0.01030 M NaOH 0.05 0.04 0.03 0.06 0.03 0.03 0.02
13.40 13.35 13.34 13.36 13.61 13.50 13.52 Average at Z, = 0.010 M: 13.44 f 0.04 b z, = 0: 13.39
-AZZ/kcal,, mol-l o(AZZ/kcal,tl mol-I) 0.06727 M HCI into 0.00292 M NaOH
a
13.10 0.10 13.06 0.03 13.27 0.06 13.12 0.06 0.06727 M HCI into 0.00281 M NaOH 13.68 0.09 13.55 0.12 13.56 0.08
0.04
Average at Z, = 0.0028 M: 13.33 f 0.10 b z, = 0: 13.30 Grand average at Z, = 0: 13.35 f 0.05 0
0.08
a Standard deviation within a run. b Standard deviation of the mean among runs in the set. c Standard deviation of the mean between sets at I, = 0.
FIGURE 3. The error g(Qcd in the corrected enthaIpy change as a function of titration time t and total heat Q produced. (The point when I = 0 is t,, the time when titrant was first added.)
926
L. D. HANSEN
ET AL.
experiments although it amounted to about 25 per cent of the total change produced in a calorimetric run. An error-propagation analysis of the equations used in the calculation of the corrected enthalpy changes Q,,, at each point was made in order to identify the major sources of error and their magnitudes. The results show that the most important source of random error is in the correction for heat lost from the Dewar although errors in the titrant temperature and in the measurement of the temperature rise to any point are also significant. The results of these calculations for the results from the calorimeter used in this study are given in figure 3. Not surprisingly, the error in Q,,, increases as the titration time increases, but the nonlinear dependence on time was unexpected. Decreasing the titration time decreases the error in Q,,, but increases the error in the titrant volume added. The line in figure 3 for Q = 0 was verified by four titrations of water into water. The standard deviation of Q from zero, $-2 meal,, at 1000 s, for these four runs was in good agreement with the total values in figure 3. REFERENCES l.(a) Christensen, J. J.; Izatt, R. M.; Hansen, L. D. Proceedings of the Seventh International Conference on Coordination Chemistry, Paper 7 Fl, p. 344, Stockholm, 1962. (b) Christensen, J. J.; Hansen, L. D.; Izatt, R. M.; Partridge, J. A. “Application of High Precision Thermometric Titration Calorimetry to Several Chemical Systems”, in Microcalorimdtrie et Thermogendse Publication No. 156 from Centre National de la Recherche Scientifique, Paris, France, p. 207, 1967. (4 Christensen, J. J.; Izatt, R. M. “Thermochemistry in inorganic solution chemistry”, chapter in Techniques in Advanced Inorganic Chemistry, Day, P. ; Hill, A.; editors. John Wiley and Sons: New York, 1968. 2. Christensen, J. J.; Eatough, D. J.; Ruckman, J.; Izatt, R. M. Thermochimica Acta 1972, 3, 203.
i: 5.
Eatough, D. J.; Christensen, J. J.; Izatt, R. M. Thermochimica Actu 1972, 3, 219. Eatough, D. J.; Izatt, R. M.; Christensen, J. J. T%ermochimica Acta 1972, 3, 233. Tyrrell, H. J. V.; Beezer, A. E. Thermometric Titrimetry Chapman and Hall, Ltd.: London, 1968.
6.
7. 8. 9. 10. 11. 12.
Hansen, L. D.; Izatt, R. M.; Christensen, J. J. Chap. 1 in New Developments in Titrimetry J. Jordan; editor. Marcel Dekker: Inc.: New York, 1974. Vaughn, G. A. Thermometric and Enthalpimetric Titrimetry Van Nostrand Reinhold Co.: London, 1973. Hansen, L. D.; Izatt, R. M.; Eatough, D. J.; Jensen, T. E.; Christensen, J. J. “I. Recent advances in titration calorimetry”, in AnalyticaI Calorimetry, Porter, R. S.; Johnson, J.; editors. Plenum Press: New York, 1974, p. 7. Izatt, R. M.; Hansen, L. D.; Eatough, D. J.; Jenson, T. E.; Christensen, J. J. “II. Recent analytical applications of titration calorimetry”, ibid., p. 237. Coops, J.; Jessup, R. S.; van Nes, K. Chap. 3 in Experimental Thermochemistry, Rossini, F. D.; editor Interscience: New York, p. 30,1956. Parker, V. B. Thermal Properties of Aqueous &i-univalent Electrolytes NSRDS-NBS 2, U.S. Government Printing Office: Washington, D.C., 1965. Hansen, L. D.; Lewis, E. A. J. Chem. Thermodynamics 1971,3,35.
Heat-loss calorimeter
corrections reaction
for small isoperibolvessels a
L. D. HANSEN, T. E. JENSEN, S. MAYNE, R. M. IZATT, and J. J. CHRISTENSEN
D. J. EATOUGH,
Departments of Chemistry and Chemical Engineering, and Contribution No. 59 from the Center for Thermochemical Studies, Brigham Young University, Provo, Utah 84602, U.S.A. (Received 13 June 1974; in revisedform
16 October 1974)
A more exact method of calculating the heat loss from isoperibol-calorimeter reaction vessels is described. This method involves corrections for changes in the heat-leak constant and the external power input with changes in the calorimeter contents, and is particularly important for reaction vessels with volumes of less than 25 cm3 or with heat-leak constants greater than about 0.005 min-I. The new calculation has been tested by application to results collected with a titration calorimeter of capacity 3 cm3.
1. Introduction Titration calorimetry has proven to be an especially useful technique for the determination of AH where simultaneous reactions exist,(‘) for the “determination” of AG for reactions involving weakly interacting species, (2-4) for the identification of species present in complex reaction mixtures,(l, 5-7) and for analytical determinations.(5-7) By reducing the size of the reaction vessel, the technique can be extended to many systems where materials are rare, expensive, or available only in limited amounts.(8P 9, Reduction of the volume of solution at a given concentration means that less material need be used while the same accuracy of determination is maintained; or, reduction of volume for a given amount of solute means that more concentrated solutions can be prepared, resulting in larger temperature changes and increased accuracy. These effects arise because while the actual enthalpy change is proportional to the extent of reaction, the temperature change is inversely proportional to the volume. Although analytical thermometric titrations have been done in as little as 5 cm3 of solution for a number of years, (5-7) this paper reports the first complete study of the requirements for isoperibol-calorimetric titrations with volumes of less than 5 cm3. The heat leakage to the surroundings is proportionately much greater from small (c 25 cm3) than from large (> 25 cm3) reaction vessels and more exact equations than previously derivedC3’ are required to correct for heat loss during the experiment. This arises because the heat leak modulus is an inverse function of the volume of the reaction vessel since there is a practical limit to the reduction in size of wires, stirrer a Supported by Grants GM18816 and AM15616 from the U.S. Public Health Service, and by NSF Grant GP33536X. 61
920
ET AL,.
L. D. HANSEN
shaft, and reaction vessel walls which constitute the connections between the system and its surroundings. Heat-leak constants, IC,as measured for various sizes of Dewar vessels specially constructed to have small vaiues of rc and used in our laboratory in the past are plotted against the volume of the reaction vessel in figure 1. It can be 0.04*
0.03‘; .; 0.02\ Y
o.ol-\
.;
;
1
'1:
60
80
100
l -• 0
20
a 40
Y/cm3 FIGURE 1. Modulus of heat leak K for specially constructed glass Dewar reaction vessels as a function of Dewar vessel volume V.
seen that as the volume of the Dewar vessel decreases the value of rc becomes greater and the heat loss to the surroundings during a titration could be equal to or even exceed the isothermal enthalpy change resulting from the chemical reaction. The relative errors resulting from the assumptions previously used to correct for heat loss then become larger and a more exact description of the heat loss must be used. This paper presents the derivation of equations accurately describing heat exchange between an isoperibol-calorimeter reaction vessel and its surroundings and the testing of those equations with experimental results.
2. Experimental EQUIPMENT
The calorimeter used in this study was a Tronac head and a 3 cm3 glass Dewar reaction vessel. thermistor bridge was (114.5kO.5) mV K -l. The meter syringe driven by a stepping motor. The detail elsewhere.“’
Model 1000 equipped with a small The temperature sensitivity of the buret was a 1 cm3 Gihnont microcalorimeter has been described in
MATERIALS TRIS solutions
were prepared by mass from TRIS (Fisher, primary standard) dried at 383 K. HCl (DuPont, reagent) solutions were standardized both by pH titration against TRIS and by Mohr titration with AgNO, (Baker, reagent). NaOH (B & A, 50 per cent, carbonate free reagent) solutions were standardized against the HCI solutions by pH titration. All water was distilled, filtered through charcoal, treated with a mixed bed ion exchange column, and freshly boiled before use.
HEAT-LOSS
CORRECTIONS
FOR SMALL
VESSELS
921
CALIBRATION
The energy equivalent E of the calorimeter was determined as a function of heating rate, heating time, and volume of water. The value of E did not vary significantly with either heating rate or time, indicating that no significant amount of heat escaped through the electrical leads of the heater. A plot of E against volume of water in the Dewar vessel was linear giving further evidence that there is no significant systematic error in the measurement of E by electrical heating. The energy equivalent of the empty vessel, (0.181+ 0.009) Cal,, K-l, was obtained by subtracting the heat capacity of the water from the total energy equivalent measured at each volume and averaging the results.? MEASUREMENTS For both enthalpy-of-reaction and energy-equivalent determinations, readings were taken at either 4 or 10 s intervals, the choice of interval length depending on the total length of time the buret or heater was operating. The total time was either 6.7 or 16.5 min. From 60 to 99 readings were taken while the buret or heater was on and one-fourth of these readings were taken during both the lead and trial periods. For chemical runs, 0.25 cm3 of either 0.25 or 0.07 M HCl was titrated, respectively, into 2.69 cm3 of 0.01 or 0.003 M NaOH or TRIS solutions. The more concentrated solutions produced changes of about 270 mcalti, and the more dilute about 100 meal,, during the titrations, corresponding to temperature rises of about 0.08 and 0.03 K, respectively. Bridge unbalance potentials were measured with a digital voltmeter with a resolution of ) 1 pV.
3. Calculations ANALYSIS OF THERMOGRAMS Equations have been presented (3) for the analysis of results of calorimetric titrations which include methods for calculating: (1) the total enthalpy change produced in the reaction vessel; (2) the correction terms due to effects from non-chemical energy; (3) the effect resulting from the difference in temperature between the titrant and titrate; (4) the ‘enthalpy of dilution of the titrant and titrate; and (5) the enthalpy change from chemical reactions other than the one of interest. From these quantities the enthalpy change of the particular reactions of interest can be calculated. The equations in reference 3 are all applicable to small reaction vessels except for those used in calculating the correction terms due to effects from non-chemical energy (2 above). These effects include stirring of the solution, heat exchange between the reaction vessel and its surroundings, Joule heating by the thermistor, and evaporation of the liquid in the reaction vessel. An equation which is valid for correcting for heat losses from reaction vessels with large heat-leak constants can be derived as follows. The rates of temperature change in the Dewar during the lead, run, and trial periods (subscripts 1 and t refer to the midpoints in time for the lead and trial periods t Throughout
this paper calm = 4.184 J; M = mol dm-+.
922
L. D. HANSEN
ET
AL.
and p to any point in the run period) caused by heat exchange with the surroundings are given by:flO) (dT/dt), = P,/si+~i(Tj - T,); (1) (dT/dt), = Pp/ap+icp(Tj- TP); (2) (dT/dt), = P&+K& - TJ; (3) where P denotes the total external power inputs, E the energy equivalents, rc the cooling constants, Tj the temperature of the surroundings, and T,, TP, and Tt the temperatures in the three periods. Multiplying (l), (2), and (3) by the respective thermal equivalent in each region and using the convention that heat input to the calorimeter is given a negative sign gives for the rates of heat exchange qHL: = -P,-slrcl(Tj-Q; qHL,I = -(dT/dt)ie, (4) -(dT/dt),s, = -P,-&pl~p(Tj-TTp); qHL,p = (5) 4 HL,t = -(dT/dt),s, = -Pt-&Tj-ZJ. (6) At this point in previous derivationst3) it has been assumed that TX = T,, Ty = T,, lcl = rcP = rcc,,PI = P, = P,, and E,I$ = ERIC,, = E,IC,, where subscripts x and y refer to times when titrant first enters and when the buret is turned off, and using these assumptions (4), (5), and (6) have been combined to obtain an equation for qm, P.(3) These simplifying assumptions are sufficient for Dewar vessels larger than 25 cm3 and with values of K less than about 0.005 mine1 in which the correction for heat exchange with the surroundings is a small fraction of the total change measured. For the Dewar vessel used in this study (3 cm3 volume, rc = 0.046 min-I). The above assumptions resulted in about 10 per cent error, nine-tenths of which arose from ElSSUUllIlg
EIJC,
=
EplCp
=
E,IC,.
In order to solve equations (4), (5), and (6) for reaction-vessel volumes less than 25 cm3 it is necessary to know how rc varies with volume. The dependence of K on volume can be determined by making electrical calibrations at different volumes. Since the volume of the liquid in the Dewar does not change during a calibration run, PI = P,, Ed = E,, rci = K~,and equations (1) and (3) may be combined and solved for K to give K = ((dT/dt),-(dT/dt),}/(T,TJ. (7) A plot of K against volume of water in the 3 cm3 Dewar vessel used in this study is shown in figure 2 and may be approximated as a straight line up to 3 cm3 at which point the value of IC rises rapidly. (*) For simplicity it is better to work at volumes included in the linear region. We thus write equations (8) and (9) for a titration performed within this linear region: Kp = 7c,+au,; (8) 7c,= Jc*+cw,; (9) where 0 is the volume of titrant added and cxthe rate of change of K with ZJ. Combining equations (4), (5), and (6) with (8) and (9) gives: ~HL,I
(10)
=-~I-EIKI(T~-Q
4 HI,, p = - p, - E&G &IL,
t =
-
p,
-
@I
+ @u,)(Tj + Nvj
- T,), -
m
(11)
(12)
HEAT-LOSS CORRECTIONS 0.150,
I
I
I
0.125 -7.c: o.loo _ d&dY = 0.0099 mine1cme3 ,I E >
923
FOR SMALL VESSELS I P I _ ;’
*’
0.075.a’
\
0.050 o.0252.0
4-a
-M
9’
”
2.5
3.0
4.0
3.5I
V/cm3 FIGURE 2. Modulus of heat leak K for a Dewar vessel, volume 3 cm3, as a function of the volume V of water in the vessel.
which contain six quantities, P,, P,, Pt, IC,, Tj, and &u, p, the values of which are not available from the experimental results collected during a titration run. Values must be assigned to three of the quantities to allow calculation of qHL, p. Tj may be approximated by setting it equal to the bath temperature. This is an approximation because there are thermal connections to the liquid in the Dewar vessel which may not be at bath temperature, i.e. the stirrer shaft and the electrical leads. (Alternatively an absolute value of rcclcould be determined from the results of electrical-heating calibration runs; however, it was found that slightly more precise values of AH were obtained using the Tj approximation.) The value of P at a given total liquid volume in the Dewar vessel may be found by plotting qHL, , against (q - IQ. The intercept at (q - T) = 0 is -P (see equation 10). By repeating these calculations at several different volumes of liquid in the reaction vessel it is possible to determine how P varies with the liquid volume. For the calorimeter used in this study P was found to be a linear function of volume@) so we write : p, = p, + PQ’ (13) P, = P, + ~~*, (14) where p is the rate of change of P with the volume of titrant v, and where equations (13) and (14) give the change in external power input as the total volume of liquid in the Dewar changes. The expression for qHL,p in terms of experimentally measurable quantities is obtained by combining equations (IO), (1 l), and (12) with (13) and (14), defining 8, = (Tj - If’,), 8, = (Tj - T,), and 8, = (Tj - T,), and eliminating P, and rcr: qHL,
p =
@HL,
I -
&,
-
Epf=Jp +
ikpep
0,) -
&1 6hwt
-
+ fmkHL,
E,, and E, can be expressed as linear functions of substitution of EP= %fPp, Et = -%+I%
t -
Ed
qHL,
1 +b+
Et~~,4).
(15)
and ZI (see section 2), but the ‘36)
(17)
924
L. D. HANSEN
ET AL.
where y is the rate of change of E with o, into (15) need not be done since in actual application numerical values of ep and at are obtained and substituted. The total heat exchange Qmp between the reaction vessel and the surroundings from the beginning of the titration toppoint p can be calculated from the equation: tP
QHL,p"
s tx
(18)
qHL dt.
The procedures used for this integration have been described in detail.(3) All other calculations involved in the analysis of the thermograms were the same as previously described.(3) CALCULATION
OF AH
Values of AH for the protonation of TRIS and the ionization of water were obtained by a linear least-squares fit of the corrected enthalpy change Qc,, with the amount of titrant added at selected points in the thermogram.(3) The points selected were those in the middle two-thirds of the reaction region. These were selected in order to avoid the problem of reaction of the titrant with any carbonate or hydrogen carbonate ions present in the solutions. Corrections for the enthalpy of dilution of the titrant were made using data from reference 11. All calculations were done on an IBM 7030 computer and FORTRAN programs are available.
4. Results and discussion of
The final results, -(11.35+0.05) TRIS (18 runs) and (13.35F0.05)
TABLE
kcal, mol-l for the enthalpy of protonation kcal,, mol -I for the enthalpy of ionization of
1. The enthalpy of protonation of TRIS with HCl at 298.15 K as determined in a 3 cm3 isoperibol titration calorimeter (cal$,, = 4.184 J; M = mol dmP3) -AH/k&,, mol-l o(AH/k&, mol-I) 0.2581 M HCl into 0.00919 M TRIS 11.24 0.01 11.17 0.01 11.28 0.01 11.32 0.01 11.19 0.02 11.22 0.01 0.2581 M HCI into 0.00903 M TRIS 11.34 0.09 11.46 0.11 11.24 0.06 11.29 0.08
Average at I, = 0.009 M: 11.28 f 0.03 b
a
-AH/k&,, mol-1 o(AH/kcalth mol-I) 0.06727 M HCl into 0.00313 M TRIS 11.73 11.14 11.44 11.05 11.31 11.55 11.64 11.68 Average at I, = 0.003 M: 11.44 f 0.09 b
0.04 Grand average: 11.35 i 0.05 c
DStandard deviation within a run. b Standard deviation of the mean among runs in the set. c Standard deviation of the mean among all runs.
0.05 0.03 0.02 0.05 0.05 0.08 0.09 0.08 0.06
a
HEAT-LOSS
CORRECTIONS
FOR SMALL
925
VESSELS
Hz0 (14 runs), are in agreement with accepted values (i.e. - 11.32 and 13.35 kcal,, mol-l, respectively).(11,‘2) The deviations both within a run and among the runs indicated that the overall accuracy of the AH obtained in this study is about +O. 1 kcal,, mol-I. (See tables 1 and 2 for a list of the values of AH and the standard deviations obtained.) The agreement between the accepted values and the values obtained in this study show that we have successfully described the heat loss from the Dewar vessel in these TABLE
2. The enthalpy of ionization of water as determined in a 3 cm3 isoperibol titration calorimeter (Cal,, = 4.184 J; M = mol dmm3)
- AZZ/kcaltb moi - 1 o(AZZ/kcalu, mol -I) a 0.2581 M HCI into 0.01030 M NaOH 0.05 0.04 0.03 0.06 0.03 0.03 0.02
13.40 13.35 13.34 13.36 13.61 13.50 13.52 Average at Z, = 0.010 M: 13.44 f 0.04 b z, = 0: 13.39
-AZZ/kcal,, mol-l o(AZZ/kcal,tl mol-I) 0.06727 M HCI into 0.00292 M NaOH
a
13.10 0.10 13.06 0.03 13.27 0.06 13.12 0.06 0.06727 M HCI into 0.00281 M NaOH 13.68 0.09 13.55 0.12 13.56 0.08
0.04
Average at Z, = 0.0028 M: 13.33 f 0.10 b z, = 0: 13.30 Grand average at Z, = 0: 13.35 f 0.05 0
0.08
a Standard deviation within a run. b Standard deviation of the mean among runs in the set. c Standard deviation of the mean between sets at I, = 0.
FIGURE 3. The error g(Qcd in the corrected enthaIpy change as a function of titration time t and total heat Q produced. (The point when I = 0 is t,, the time when titrant was first added.)
926
L. D. HANSEN
ET AL.
experiments although it amounted to about 25 per cent of the total change produced in a calorimetric run. An error-propagation analysis of the equations used in the calculation of the corrected enthalpy changes Q,,, at each point was made in order to identify the major sources of error and their magnitudes. The results show that the most important source of random error is in the correction for heat lost from the Dewar although errors in the titrant temperature and in the measurement of the temperature rise to any point are also significant. The results of these calculations for the results from the calorimeter used in this study are given in figure 3. Not surprisingly, the error in Q,,, increases as the titration time increases, but the nonlinear dependence on time was unexpected. Decreasing the titration time decreases the error in Q,,, but increases the error in the titrant volume added. The line in figure 3 for Q = 0 was verified by four titrations of water into water. The standard deviation of Q from zero, $-2 meal,, at 1000 s, for these four runs was in good agreement with the total values in figure 3. REFERENCES l.(a) Christensen, J. J.; Izatt, R. M.; Hansen, L. D. Proceedings of the Seventh International Conference on Coordination Chemistry, Paper 7 Fl, p. 344, Stockholm, 1962. (b) Christensen, J. J.; Hansen, L. D.; Izatt, R. M.; Partridge, J. A. “Application of High Precision Thermometric Titration Calorimetry to Several Chemical Systems”, in Microcalorimdtrie et Thermogendse Publication No. 156 from Centre National de la Recherche Scientifique, Paris, France, p. 207, 1967. (4 Christensen, J. J.; Izatt, R. M. “Thermochemistry in inorganic solution chemistry”, chapter in Techniques in Advanced Inorganic Chemistry, Day, P. ; Hill, A.; editors. John Wiley and Sons: New York, 1968. 2. Christensen, J. J.; Eatough, D. J.; Ruckman, J.; Izatt, R. M. Thermochimica Acta 1972, 3, 203.
i: 5.
Eatough, D. J.; Christensen, J. J.; Izatt, R. M. Thermochimica Actu 1972, 3, 219. Eatough, D. J.; Izatt, R. M.; Christensen, J. J. T%ermochimica Acta 1972, 3, 233. Tyrrell, H. J. V.; Beezer, A. E. Thermometric Titrimetry Chapman and Hall, Ltd.: London, 1968.
6.
7. 8. 9. 10. 11. 12.
Hansen, L. D.; Izatt, R. M.; Christensen, J. J. Chap. 1 in New Developments in Titrimetry J. Jordan; editor. Marcel Dekker: Inc.: New York, 1974. Vaughn, G. A. Thermometric and Enthalpimetric Titrimetry Van Nostrand Reinhold Co.: London, 1973. Hansen, L. D.; Izatt, R. M.; Eatough, D. J.; Jensen, T. E.; Christensen, J. J. “I. Recent advances in titration calorimetry”, in AnalyticaI Calorimetry, Porter, R. S.; Johnson, J.; editors. Plenum Press: New York, 1974, p. 7. Izatt, R. M.; Hansen, L. D.; Eatough, D. J.; Jenson, T. E.; Christensen, J. J. “II. Recent analytical applications of titration calorimetry”, ibid., p. 237. Coops, J.; Jessup, R. S.; van Nes, K. Chap. 3 in Experimental Thermochemistry, Rossini, F. D.; editor Interscience: New York, p. 30,1956. Parker, V. B. Thermal Properties of Aqueous &i-univalent Electrolytes NSRDS-NBS 2, U.S. Government Printing Office: Washington, D.C., 1965. Hansen, L. D.; Lewis, E. A. J. Chem. Thermodynamics 1971,3,35.