1.12.1: Heat Capacities- Isobaric- Solutions

From the definition of enthalpy \(\mathrm{H}\), an infinitesimal small change in enthalpy is related to the corresponding change in thermodynamic energy \(\mathrm{dU}\) by equation (a).

\[\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]

If only ‘\(\mathrm{p}-\mathrm{V}\)’ work is involved,

\[\mathrm{dU}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}\]

Then

\[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}\]

But in general terms,

\[\mathrm{q}=\mathrm{C} \, \mathrm{dT}\]

Here \(\mathrm{C}\) is the heat capacity of the system, an extensive variable. Hence for a change at constant pressure,

\[\mathrm{dH}=\mathrm{C}_{\mathrm{p}} \, \mathrm{dT}\]

Isobaric heat capacity is related to the change in enthalpy accompanying a change in temperature.

\[\mathrm{C}_{p}=(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}\]

\(\mathrm{C}_{\mathrm{p}}\) is an extensive variable; \(\mathrm{C}_{\mathrm{pm}}\) is the corresponding molar property. We develop the above analysis in a slightly different way in order to make an important point. We explore the relationship between the dependence of (\(\mathrm{G}/\mathrm{T}\)) on temperature at

1. constant affinity and
2. constant extent of reaction.

A calculus operation yields the following equation.

\[\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]

But at equilibrium,

\[A=-\left[\frac{\partial G}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}=0\]

Using the Gibbs-Helmholtz Equation,

\[\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\]

This result is expected because enthalpy \(\mathrm{H}\) is a strong state variable, a function of state which does not need a description of a pathway. This is not the case for isobaric heat capacities. Using the same calculus operation,

\[\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]

We cannot assume that the triple product term is zero. Hence there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) and the ‘frozen’ isobaric heat capacity, \(\mathrm{C}_{\mathrm{p}\left(\xi_{\mathrm{eq}\right)\).

\[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]

In other words, the isobaric heat capacity is not a strong function of state. The property is concerned with a pathway between states. The term \left[-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\right]\) is the relaxational isobaric heat capacity. \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\), the equilibrium heat capacity, signals that when heat q passes into a system, the composition - organization of the system changes in order that the Gibbs energy of the system remains at a minimum. In contrast \(\mathrm{C}_{\mathrm{p}}\left(\xi_{\mathrm{eq}}\right)\), the frozen heat capacity, signals that no changes occur in the composition – organization in the system such that the Gibbs energy is displaced from the original minimum. Moreover the equilibrium isobaric heat capacity is always larger than the frozen isobaric heat capacity. Indeed we can often treat the extensive equilibrium property \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) as a function of state.

Certainly isobaric heat capacities differentiate water as a solvent from other associated liquids [1,2] such as \(\mathrm{H}_{2}\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\mathrm{H}_{4}\) and low melting fused salts such as ethylammonium nitrate. Interestingly, among liquids, water has one of the highest heat capacitances; i.e. heat capacities per unit volume [3]. Therefore hypothermia is often life threatening for babies and old persons because in order to raise their temperature a large amount of thermal energy has to be passed into the body in order to raise their temperature. This is often difficult without damaging the skin and other body tissues--- a consequence of humans being effectively concentrated aqueous systems.

The isobaric heat capacity of a solution prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is defined by equation (1).

\[\mathrm{C}_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]

We assume the system is at thermodynamic equilibrium such that the affinity for spontaneous change is zero at a minimum in Gibbs energy. The isobaric heat capacity of the solution is related to the composition using equation (m).

\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{pl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\]

Here \(\mathrm{C}_{\mathrm{p}1}(\mathrm{aq})\) and \(\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\) are the partial molar isobaric heat capacities enthalpies of solvent and solute respectively. Alternatively \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq})\) is given by equation (n) where \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\) and \(\phi\left(\mathrm{C}_{\mathrm{pj}}\right)\) are the molar isobaric heat capacity of the pure solvent and the apparent molar isobaric heat capacity of the solute \(j\) respectively. Thus

\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]

Footnotes

[1] M. Allen, D. F. Evans and R. Lumry, J. Solution Chem.,1985,14,549.

[2] M. Hadded, M. Biquard, P. Letellier and R. Schaal, Can. J. Chem.,1985,63,565.

[3]

Liquid	Heat Capacitance \(\mathrm{C}_{\mathrm{p}} / \mathrm{~J K}^{-1} \mathrm{~cm}^{-3}\)
Water	4.18
Propane	1.67
Cyanomethane	2.26
Ethanol	1.92
Tetrachloroethane	1.38

See J. K. Grime, in Analytical Solution Calorimetry, ed. J. K.Grime, Wiley, New York, 1985, chapter 1.

For isochoric and isobaric heat capacities of liquids see, D. Harrison and E. A. Moelwyn-Hughes, Proc. R. Soc. London, Ser.A,1957, 239, 230.