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On the theory of thermometric titration
SHOUT T~NIIIO. Vol. 21. pp. 975. 97X. Pergamon
Press. 1974. Printed
ON THE THEORY
975
COMMUNICATIONS
in Great Brnain
OF THERMOMETRIC
TITRATION
(Received 18 December 1973. Accepted 8 February 1974) In recent years, the thermometric titration method has become extensively used in analytical and physico-chemical laboratories. Many aspects of the theory have been c1arified,lw5 yet there remains the question of the role of heat emission during the thermometric measurements. That heat transfer between ambient medium and thermometric cell has an important effect on the measurements has been confirmed many times(cf’. Tyrrell and Beezer’). However, in the formulae used for thermometric titration calculations this factor is, unfortunately, either not accounted fo?.’ or the formulae are complicated to such an extent as to become unsuitable for practical calculations. 4*5*7This paper is an attempt to fill this gap. Thermometric titration yields a curve which is plotted in terms of solution temperature T VS.time t, or T us. volume 1’of titrant added. Such curves are equivalent to each other at a constant titration rate. Figure 1 represents a general curve, which may be subdivided into three sections. The first corresponds to the period before titration is started. In this period, heat sources are friction during mixing, and heat transfer with the ambient medium. The second section is the main one, and the additional heat sources are the chemical reaction and mixing or dilution of the solution. The third section represents the final step of the experiment. The reaction is completed, but the titrant continues to flow into the solution at a constant rate. The heat sources are the same as those in the second section. with the exception of the chemical reaction.
i.$3
fk vk
t
NC
V cm=
Fig. 1. Thermometric titration curve to(V,) = start of titration, tt( Vk)= end-point.
Only the second step is of interest to us here. We assume that the titrant is added at constant rate, that heat transfer between the solution and the ambient medium occurs in accordance with Newton’s law, with a constant heat-transfer coefficient, that mixing is so efficient that temperature and concentration are uniform throughout the solution, that the temperature-dependence of the specific heat capacity, heat-transfer coefficient and heats of mixing and reaction may be neglected, that frictional heat is generated at a constant rate of w cal/sec, that the solution composition and ionic strength do not influence the results, and that the reaction rate is infinitely high. To make the argument general. the initial temperature of the solution r,, (at time to = O),temperature of titrant r,. and temperature of the ambient medium T, are taken as not equal to one another. We consider reactions of the type mA + nB -+ A,B, (1) where B is the titrant.
976
SHORT
COMMUNlCATlONS
The thermal balance of the thermometric cell may be expressed as:* a(M,C,+hGY)~+q+lF(T-T,)+p,C,a(T-T,)=O
(2)
where o is the titration rate (cm”secl III,,and C,, are the mass (g) and the specific heat capacity (cat .g- ’ .deg- ‘) respectively of the thermometric cell + starting solution system or and C, are the density (g/cm3) and the specific heat capacity respectively of the titrant (cal .g- ’ . deg- ‘); V is the volume of titrant (cm3) added in time r. x is the heat-transfer coefficient (cal .cm-* . deg- ’ . set- ‘); F is the surface area for heat-transfer (cm2), 4 is the rate of production of internal heat: 4 = Q~Gamln + QVPT~ + wlp,a (3) where Qa is the heat ofchemical reaction @al/mole); Ca is the concentration of the titrant (mole/cm3). Qv is the heat of mixing (Cal/g of titrant), and T is the solution temperature at time t. The initial condition is that V = 0 and T= T,,. Equation (2) can be solved in a conventional manner* to yield 1 + To - T, + K,Vo - TT)
AT=T-T,,i
(4) where 1= q/rF, K, = m,C,a/zF and Kz = pTCTa/zF. Sina K, V/K, 4 1, then to a good approximation
Equation (4) may then be written as AV 1+ BY
AT=T-T,,=-_
(5)
where A
=
1+ TO- Tc+ K,(To
-
T,),
B=
K,
1 +
K,
K,
’
both being constants. Since usually 1 $DTo - T, and 1 p K2(To - TT) the sign of AT is determined by the sign of q. By comparing equations (2) and (5) with those in the literature, it can be proved that the Keily and Hume equation6 is a special case ofequation (2). Although the thermal-balance equations proposed by other&’ agree in general with equation (2) except in some details, the authors cited could not find a concise and simple solution of their equations. Equation (5) represents the experimental data satisfactorily. Table 1 presents as an example the experimental and calculated data from Keily and Hume’s work.6 Comparison with the results calculated by the use of (5) shows that the deviation between the experimental and calculated values is an order of magnitude smaller when equation (5) is used. Table 1. Comparison between the experimental and calculated values of the temperature change (“C) in titration of sodium acetate with perchloric acid (in glacial acetic acid as medium) I’. III/of 0.5M 0.5 1.50 2.50 3.50 4.50 5.50 5.95
HCIO,
AT (exp.)
AT, (theor.)*
AT2 (theor.)t
AT - AT,
AT - AT2
0.045 0.134 0213 0.285 0353 0.418 0442
0.047 @I36 0.225. 0.314 0.399 0483 0521
0.045 0.131 0.21 I 0.285 0.355 0.42 1 0449
- oXtO - 0002 -0.012 -0-029 -0046 -0.065 - 0.079
0 + 0.003 + 0.002 0 - 0.002 -@CO3 -o+Io7
* Calculations according to Keily and Hume.6 t Calculations according to equation (5). A = -0092,
B = 0.0368.
* We have chosen the titrant volume V as an independent variable, because it is directly proportional titration time. as indicated by the assumed condition a = du/dt = const.
to the
977
SHORTCOMMUNICATIONS
The most interesting conclusion which may be drawn from (5) is that in the general case the value of AT is not directly proportional to I’. as is sometimes assumed to be the case. The linear dependence should exist between the inverse values: I !A7 and I ‘1’. Consequcntl\. it would he Ibct~cr IO plot not AT I‘.\. I the usual practice. but l/AT I‘.,. lit’, which should give a straight hne. This can easily be proved if (5) is written in the form 1
1
B
(1
AV
A
V
-__-_=---/J
rT=
where a = l/A = const.. h = B/o = const. In these graphs. the point of inflexion corresponding
to the end-point should become more conspicuous,
The Insritute,{cr Geoloyy of Ore Deposits. Petrography, Mirleralog_v and Geochemisrr~ U.S.S.R. Acadent!, of Sciences Staronionetny 35. Moscow 10901 I. U.S.S.R.
1. 2. 3. 4. 5. 6. 7. 8.
G.O. PILOYAN
Yu.V.
DOLININA
REFERENCES H. J. V. Tyrrell and A. E. Beezer, Thermometric Tirrimetry. Chapman and Hall, London, 1968. L. S. Bark and S. M. Bark, Thermometric Tinimetr); Pergamon. London, 1969. J. Jordan and T. G. Alleman, Anal. Chem., 1957, 29, 9. J. J. Christensen. R. M. Izatt. L. D. Hansen and J. A. Partridge. J. Phys. Chem.. 1966, 70, 2003. J. Barthel. F. Becker and N. G. Schmahl. 2. Physik. Chem. (Frankfurr). 1961, 29, 58. H. J. Keily and D. N. Hume, Anal. Chem., 1956, 28, 1294. J. Barthel and N. G. Schmahl. Z. AMal. Chem., 1968, 233, 328. E. Kamke. D$@erlrial Eyuarions. 2nd Ed.. State Publishing House of Foreign Literature, Moscow, 1951,
Summary-The
thermometric
general equation titration is
defining the change in solution temperature
AT during a
AV
AT=T-T,,=_---
1 + BV
where A and B are constants, V is the volume of titrant used to produce temperature T, and T,, is the initial temperature. There is a linear relation between the inverse values of AT and V:
1
a
AT
V
__=---b
where 11= 1/A and b = B/A. both a and h being constants. A linear relation between AT and V is usually a special case of this general relation. and is valid only over a narrow range of V. Graphs of 1/AT cs. I W are more suitable for practical calculations than the usual graphs of AT vs. V.
aljgemeine Gleichung. die die &nderung wahrend einer thermometrtschen Titration angibt, lautet
Zusammenfassung-Die
der Losungstemperatur
AT
AC
AT=T-T,=-_
I + B1’
wo A und B Konstanten sind, V das Titrantvolumen, das die Temperatur T herbeifiihrt, und Te die Anfangstemperatur bedeuten. Es besteht eine lineare Beziehung zwischen den reziproken Werten von AT und V: I 7=-1-
0
h
wo a = 1i’A und b = B/A ebenfalls konstant sind. Eine lineare Beziehung zwischen AT und V ist im allgemeinen ein Spezialfall dieser allgemeinen Beziehung und gilt nu; in einem kleinen Bereich von 6’. Graphische Darstellungen von 1/AT gegen 1/V sind fiir Berechnungen in der Praxis besser geeignet als die iiblichen graph&hen Darstellungen von AT gegen V.
978
SHORT
COMMUNICATIONS
RkumCL’equation generale dtfinissant la variation de la temperature de solution AT durant un titrage thermometrique est: AV
AT=T-T,,=_-
I + BV
ori A et B sont des constantes, V est le volume d’agent de titrage utilise pour produire la temperature T, et T,, est la temperature initiale. 11y a une relation lineaire entre les valeurs inverses de ATet V:
-=1 AT
a
-_-
b
V
oti a = l/A et b = B/A, a et b etant toutes deux des constantes. Une relation lineaire entre AT et V est habituellement un cas special de cette relation g&kale. et est valable seulement dans un ttroit domaine de V. Les graphiques de l/AT par rapport a I/V sont plus convenables pour les cakuls pratiques que les graphiques habituels de AT par rapport a V.
Press. 1974. Printed
ON THE THEORY
975
COMMUNICATIONS
in Great Brnain
OF THERMOMETRIC
TITRATION
(Received 18 December 1973. Accepted 8 February 1974) In recent years, the thermometric titration method has become extensively used in analytical and physico-chemical laboratories. Many aspects of the theory have been c1arified,lw5 yet there remains the question of the role of heat emission during the thermometric measurements. That heat transfer between ambient medium and thermometric cell has an important effect on the measurements has been confirmed many times(cf’. Tyrrell and Beezer’). However, in the formulae used for thermometric titration calculations this factor is, unfortunately, either not accounted fo?.’ or the formulae are complicated to such an extent as to become unsuitable for practical calculations. 4*5*7This paper is an attempt to fill this gap. Thermometric titration yields a curve which is plotted in terms of solution temperature T VS.time t, or T us. volume 1’of titrant added. Such curves are equivalent to each other at a constant titration rate. Figure 1 represents a general curve, which may be subdivided into three sections. The first corresponds to the period before titration is started. In this period, heat sources are friction during mixing, and heat transfer with the ambient medium. The second section is the main one, and the additional heat sources are the chemical reaction and mixing or dilution of the solution. The third section represents the final step of the experiment. The reaction is completed, but the titrant continues to flow into the solution at a constant rate. The heat sources are the same as those in the second section. with the exception of the chemical reaction.
i.$3
fk vk
t
NC
V cm=
Fig. 1. Thermometric titration curve to(V,) = start of titration, tt( Vk)= end-point.
Only the second step is of interest to us here. We assume that the titrant is added at constant rate, that heat transfer between the solution and the ambient medium occurs in accordance with Newton’s law, with a constant heat-transfer coefficient, that mixing is so efficient that temperature and concentration are uniform throughout the solution, that the temperature-dependence of the specific heat capacity, heat-transfer coefficient and heats of mixing and reaction may be neglected, that frictional heat is generated at a constant rate of w cal/sec, that the solution composition and ionic strength do not influence the results, and that the reaction rate is infinitely high. To make the argument general. the initial temperature of the solution r,, (at time to = O),temperature of titrant r,. and temperature of the ambient medium T, are taken as not equal to one another. We consider reactions of the type mA + nB -+ A,B, (1) where B is the titrant.
976
SHORT
COMMUNlCATlONS
The thermal balance of the thermometric cell may be expressed as:* a(M,C,+hGY)~+q+lF(T-T,)+p,C,a(T-T,)=O
(2)
where o is the titration rate (cm”secl III,,and C,, are the mass (g) and the specific heat capacity (cat .g- ’ .deg- ‘) respectively of the thermometric cell + starting solution system or and C, are the density (g/cm3) and the specific heat capacity respectively of the titrant (cal .g- ’ . deg- ‘); V is the volume of titrant (cm3) added in time r. x is the heat-transfer coefficient (cal .cm-* . deg- ’ . set- ‘); F is the surface area for heat-transfer (cm2), 4 is the rate of production of internal heat: 4 = Q~Gamln + QVPT~ + wlp,a (3) where Qa is the heat ofchemical reaction @al/mole); Ca is the concentration of the titrant (mole/cm3). Qv is the heat of mixing (Cal/g of titrant), and T is the solution temperature at time t. The initial condition is that V = 0 and T= T,,. Equation (2) can be solved in a conventional manner* to yield 1 + To - T, + K,Vo - TT)
AT=T-T,,i
(4) where 1= q/rF, K, = m,C,a/zF and Kz = pTCTa/zF. Sina K, V/K, 4 1, then to a good approximation
Equation (4) may then be written as AV 1+ BY
AT=T-T,,=-_
(5)
where A
=
1+ TO- Tc+ K,(To
-
T,),
B=
K,
1 +
K,
K,
’
both being constants. Since usually 1 $DTo - T, and 1 p K2(To - TT) the sign of AT is determined by the sign of q. By comparing equations (2) and (5) with those in the literature, it can be proved that the Keily and Hume equation6 is a special case ofequation (2). Although the thermal-balance equations proposed by other&’ agree in general with equation (2) except in some details, the authors cited could not find a concise and simple solution of their equations. Equation (5) represents the experimental data satisfactorily. Table 1 presents as an example the experimental and calculated data from Keily and Hume’s work.6 Comparison with the results calculated by the use of (5) shows that the deviation between the experimental and calculated values is an order of magnitude smaller when equation (5) is used. Table 1. Comparison between the experimental and calculated values of the temperature change (“C) in titration of sodium acetate with perchloric acid (in glacial acetic acid as medium) I’. III/of 0.5M 0.5 1.50 2.50 3.50 4.50 5.50 5.95
HCIO,
AT (exp.)
AT, (theor.)*
AT2 (theor.)t
AT - AT,
AT - AT2
0.045 0.134 0213 0.285 0353 0.418 0442
0.047 @I36 0.225. 0.314 0.399 0483 0521
0.045 0.131 0.21 I 0.285 0.355 0.42 1 0449
- oXtO - 0002 -0.012 -0-029 -0046 -0.065 - 0.079
0 + 0.003 + 0.002 0 - 0.002 -@CO3 -o+Io7
* Calculations according to Keily and Hume.6 t Calculations according to equation (5). A = -0092,
B = 0.0368.
* We have chosen the titrant volume V as an independent variable, because it is directly proportional titration time. as indicated by the assumed condition a = du/dt = const.
to the
977
SHORTCOMMUNICATIONS
The most interesting conclusion which may be drawn from (5) is that in the general case the value of AT is not directly proportional to I’. as is sometimes assumed to be the case. The linear dependence should exist between the inverse values: I !A7 and I ‘1’. Consequcntl\. it would he Ibct~cr IO plot not AT I‘.\. I the usual practice. but l/AT I‘.,. lit’, which should give a straight hne. This can easily be proved if (5) is written in the form 1
1
B
(1
AV
A
V
-__-_=---/J
rT=
where a = l/A = const.. h = B/o = const. In these graphs. the point of inflexion corresponding
to the end-point should become more conspicuous,
The Insritute,{cr Geoloyy of Ore Deposits. Petrography, Mirleralog_v and Geochemisrr~ U.S.S.R. Acadent!, of Sciences Staronionetny 35. Moscow 10901 I. U.S.S.R.
1. 2. 3. 4. 5. 6. 7. 8.
G.O. PILOYAN
Yu.V.
DOLININA
REFERENCES H. J. V. Tyrrell and A. E. Beezer, Thermometric Tirrimetry. Chapman and Hall, London, 1968. L. S. Bark and S. M. Bark, Thermometric Tinimetr); Pergamon. London, 1969. J. Jordan and T. G. Alleman, Anal. Chem., 1957, 29, 9. J. J. Christensen. R. M. Izatt. L. D. Hansen and J. A. Partridge. J. Phys. Chem.. 1966, 70, 2003. J. Barthel. F. Becker and N. G. Schmahl. 2. Physik. Chem. (Frankfurr). 1961, 29, 58. H. J. Keily and D. N. Hume, Anal. Chem., 1956, 28, 1294. J. Barthel and N. G. Schmahl. Z. AMal. Chem., 1968, 233, 328. E. Kamke. D$@erlrial Eyuarions. 2nd Ed.. State Publishing House of Foreign Literature, Moscow, 1951,
Summary-The
thermometric
general equation titration is
defining the change in solution temperature
AT during a
AV
AT=T-T,,=_---
1 + BV
where A and B are constants, V is the volume of titrant used to produce temperature T, and T,, is the initial temperature. There is a linear relation between the inverse values of AT and V:
1
a
AT
V
__=---b
where 11= 1/A and b = B/A. both a and h being constants. A linear relation between AT and V is usually a special case of this general relation. and is valid only over a narrow range of V. Graphs of 1/AT cs. I W are more suitable for practical calculations than the usual graphs of AT vs. V.
aljgemeine Gleichung. die die &nderung wahrend einer thermometrtschen Titration angibt, lautet
Zusammenfassung-Die
der Losungstemperatur
AT
AC
AT=T-T,=-_
I + B1’
wo A und B Konstanten sind, V das Titrantvolumen, das die Temperatur T herbeifiihrt, und Te die Anfangstemperatur bedeuten. Es besteht eine lineare Beziehung zwischen den reziproken Werten von AT und V: I 7=-1-
0
h
wo a = 1i’A und b = B/A ebenfalls konstant sind. Eine lineare Beziehung zwischen AT und V ist im allgemeinen ein Spezialfall dieser allgemeinen Beziehung und gilt nu; in einem kleinen Bereich von 6’. Graphische Darstellungen von 1/AT gegen 1/V sind fiir Berechnungen in der Praxis besser geeignet als die iiblichen graph&hen Darstellungen von AT gegen V.
978
SHORT
COMMUNICATIONS
RkumCL’equation generale dtfinissant la variation de la temperature de solution AT durant un titrage thermometrique est: AV
AT=T-T,,=_-
I + BV
ori A et B sont des constantes, V est le volume d’agent de titrage utilise pour produire la temperature T, et T,, est la temperature initiale. 11y a une relation lineaire entre les valeurs inverses de ATet V:
-=1 AT
a
-_-
b
V
oti a = l/A et b = B/A, a et b etant toutes deux des constantes. Une relation lineaire entre AT et V est habituellement un cas special de cette relation g&kale. et est valable seulement dans un ttroit domaine de V. Les graphiques de l/AT par rapport a I/V sont plus convenables pour les cakuls pratiques que les graphiques habituels de AT par rapport a V.