1.3.4: Calorimetry- Solutions- Adiabatic
A general equation describes heat \(q\) in terms of changes in temperature and composition at constant pressure; \(\mathrm{dH} = q\).
\[\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{a} \]
In this application of Equation \ref{a}, the system is thermally insulated; i.e. \(q\) is zero. An aliquot of solution containing a small amount of chemical substance \(j\) is added to a solution held in a thermally insulated container. A rearranged Equation \ref{a} takes the following form.
\[\mathrm{dT}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \mathrm{d} \xi\label{b} \]
Chemical reaction occurs in the sample cell, the rate of chemical reaction being governed by the composition of the solution and appropriate rate constants. The differential isobaric dependence of temperature on time, \(\mathrm{dT} / \mathrm{dt}\) is given by Equation \ref{c}.
\[\frac{\mathrm{dT}}{\mathrm{dt}}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}}\label{c} \]
The calorimeter records the dependence of temperature on time. An equation based on the Law of Mass Action yields the rate of change of composition \(\mathrm{d}\xi / \mathrm{dt}\). The integrated form of Equation \ref{c} yields a calculated dependence of \(\mathrm{T}\) on time which can be compared with the recorded dependence. This subject is important in the context of thermal imaging calorimetry [1-4].
Footnotes
[1] M. J. Blandamer, P. M. Cullis, and P. T. Gleeson, Phys. Chem. Chem. Phys.,2002, 4 ,765.
[2] B. Jandeleit, D. J. Schaefer, T. S. Powers, H. W. Turner and W. H. Weinbereg, Angew. Chem. Int. Ed. Engl.,1999, 38 ,2495.
[3] M. T. Reetz, M. H. Becker, K. M. Kuling and A. Holzwarth, Angew. Chem. Int. Ed. Engl.,1998, 37 ,2647.
[4] G. C. Davies, R. S. Hutton, N. Millot, S. J. F. Macdonald, M. S. Hansom and I. B. Campbell, Phys. Chem. Chem.Phys.,2002, 4 ,1791.