1.12.6: Heat Capacities- Isobaric- Dependence on Temperature
Interesting cases emerge where an equilibrium isobaric heat capacity reflects a change in composition as a consequence of the system changing composition in order to hold the system at equilibrium. The development of sensitive scanning calorimeters stimulated research in this subject [1], particularly with respect to biochemical research [2]; e.g. multilamellar systems [3].
We consider the case where a solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{\mathrm{x}}\) moles of solute \(\mathrm{X}\). In solution at temperature \(\mathrm{T}\) (and fixed pressure \(\mathrm{p}\)) the following chemical equilibrium is established.
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\[\mathrm{X}(\mathrm{aq}) \Longrightarrow \mathrm{Y}(\mathrm{aq}) \nonumber \]
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| At equilibirum | \(\mathrm{n}_{\mathrm{x}}^{0}-\xi\) | \(\xi\) | moles |
| or, | \(\mathrm{n}_{\mathrm{x}}^{0} \,\left[1-\frac{\xi}{\mathrm{n}_{\mathrm{x}}^{0}}\right]\) | \(\mathrm{n}_{\mathrm{x}}^{0} \,\left[\frac{\xi}{\mathrm{n}_{\mathrm{x}}^{0}}\right]\) | moles |
By definition \(\alpha=\xi / \mathrm{n}_{\mathrm{x}}^{0}\), the degree of reaction forming substance \(\mathrm{Y}\) at equilibrium. Then \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{x}}^{0} \,(1-\alpha)\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}=\alpha \, \mathrm{n}_{\mathrm{x}}^{0}\).
If \(\mathrm{w}_{1}\) is the mass of solvent, water(\(\ell\)), the equilibrium molalities are \(\mathrm{m}_{\mathrm{x}}^{0} \,(1-\alpha)\) for chemical substance \(\mathrm{X}\) and \(\alpha \, \mathrm{m}_{\mathrm{x}}^{0}\) for chemical substance \(\mathrm{Y}\). For the purposes of the arguments advanced here we assume that the thermodynamic properties of the solution are ideal. The equilibrium composition of the closed system at defined temperature and pressure is described by the equilibrium constant \(\mathrm{K}^{0}\). Then,
\[\mathrm{K}^{0}=\alpha /(1-\alpha) \nonumber \]
Hence the (dimensionless and intensive) degree of reaction,
\[\alpha=\mathrm{K}^{0} /\left(1+\mathrm{K}^{0}\right) \nonumber \]
Because \(\mathrm{K}^{0}\) is dependent on temperature then so is the degree of reaction. The extent to which an increase in temperature favours or disfavours formation of more \(\mathrm{Y}(\mathrm{aq})\) depends on the sign of the enthalpy of reaction, \(\Delta_{r}\mathrm{H}^{0}\). Thus [4,5],
\[\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{\mathrm{d} \ln \mathrm{K}^{0}}{\mathrm{dT}}\right] \nonumber \]
The analysis at this point is considerably simplified if we assume that the limiting enthalpy of reaction, \(\Delta_{r}\mathrm{H}^{\infty}(\mathrm{aq})\) for the chemical reaction is independent of temperature (at pressure \(\mathrm{p}\)). Hence using the van’t Hoff equation,
\[\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{1}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \nonumber \]
Thus the shift in the composition of the solution depends on the sign of the limiting enthalpy of reaction. If \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})<0\), an increase in temperature favours an increase in the amount of \(\mathrm{X}(\mathrm{aq})\) at the expense of \(\mathrm{Y}(\mathrm{aq})\). At temperature \(\mathrm{T}\), the enthalpy of the solution is given by equation (e) where \(\mathrm{H}_{1}^{*}(\ell)\) is the molar enthalpy of the solvent.
\[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \,(1-\alpha) \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{n}_{\mathrm{x}}^{0} \, \alpha \, \mathrm{H}_{\mathrm{y}}^{\infty}(\mathrm{aq}) \nonumber \]
or,
\[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{x}}^{0} \,\left[\mathrm{H}_{\mathrm{y}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})\right] \nonumber \]
or,
\[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{x}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \nonumber \]
We assume that \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\) is independent of temperature together with the amount \(\mathrm{n}_{1}\). Then,
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\left(\frac{\partial \mathrm{H}(\mathrm{aq}: \mathrm{A}=0)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Hence,
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\left\{\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{C}_{\mathrm{px}}^{\infty}(\mathrm{aq})\right\}+\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{n}_{\mathrm{x}}^{0}(\mathrm{~d} \alpha / \mathrm{dT}) \nonumber \]
The terms in the { } brackets are not (formally) dependent on temperature and constitute a frozen contribution to \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}, \mathrm{A}=0)\), \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \xi)\). Then equations (d) and (i) yield an equation for \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \mathrm{A}=0)\) in terms of \(\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}\).
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\xi: \mathrm{aq})+\left[\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{RT}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\right] \, \mathrm{n}_{\mathrm{x}}^{0} \nonumber \]
in terms of one mole of chemical substance \(\mathrm{X}\) (i.e. \(\mathrm{n}_{\mathrm{x}}^{0}=1 \mathrm{~mol}\)) [6],
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\xi ; \mathrm{aq})+\left[\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\right] \nonumber \]
According to equation (k) a large equilibrium heat capacity is favoured by a high \(\mathrm{C}_{\mathrm{p}}(\xi ; \mathrm{aq})\) and a large enthalpy of reaction. The term \(\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}\) ensures that irrespective of whether the reaction (as written) is exothermic or endothermic, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi)\) is positive. The dependence of \(\left[\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi: \mathrm{aq})\right]\) on temperature forms a bell-shaped plot covering the range of temperatures when all added substance X is completely in the form of \(\mathrm{X}\) or of \(\mathrm{Y}\). The maximum in the bell occurs near the temperature at which \(\mathrm{K}^{0}\) is unity [7]. If \(\mathrm{K}^{0}\) is unity, at this temperature \(\Delta_{r}\mathrm{G}^{0}\) is zero. In other words, at this temperature the reference chemical potentials of \(\mathrm{X}\) and \(\mathrm{Y}\), \(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) respectively are equal. Clearly therefore the temperature at the maximum in \(\mathrm{C}_{p}(\mathrm{~A}=0)-\mathrm{C}_{p}(\xi)\) is characteristic of the two solutes [8]. Equation (k) forms the basis of the technique of differential scanning calorimetry (DSC) as applied to the investigation of the thermal stability of biologically important macromolecules [2,9,10]. In the text book case, a plot of isobaric heat capacity against temperature forms a bell-shaped curve, the maximum corresponding to temperature at which the equilibrium constant for an equilibrium having the simple form discussed above is unity. The area under the curve yields the enthalpy change characterising the transition between the two forms \(\mathrm{X}\) and \(\mathrm{Y}\) of a single substance.
The possibility exists that the temperature dependences of\(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) are such that the two plots intersect at two temperatures producing two maxima in the plot of \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi)\) against temperature.
The patterns recorded by DSC scans for a \(\mathrm{X} \leftrightarrows \mathrm{Y}\) system can be understood in terms of the separate dependences of \(\left(\mu_{\mathrm{X}}^{0} / \mathrm{T}\right)\) and \(\left(\mu_{\mathrm{Y}}^{0} / \mathrm{T}\right)\) on temperature, where \(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) are the standard chemical potentials of substances \(\mathrm{X}\) and \(\mathrm{Y}\). The maximum in the recorded heat capacity occurs where the plots of \(\left(\mu_{\mathrm{X}}^{0} / \mathrm{T}\right)\) and \(\left(\mu_{\mathrm{Y}}^{0} / \mathrm{T}\right)\) against temperature cross [11,12]. If these curves have a more complicated shape there is the possibility that they will cross at two temperatures. In fact this observation raises the possibility of identifying hot and cold denaturation of proteins using DSC. Similar extrema in isobaric heat capacities are recorded for gel-to-liquid transitions in vesicles [13,14].
In more complex systems, the overall DSC scan can indicate the presence of domains in a macromolecule which undergo structural changes when the temperature is raised [15,16].
Analysis of extrema in heat capacities becomes somewhat more complicated when two or more equilibria are coupled [17,18].
Footnotes
[1] V. V. Plotnikov, J. M. Brandts, L.-N. Lin and J. F. Brandts, Anal. Biochem., 1997, 250 ,237.
[2] J. M. Sturtevant, Ann. Rev. Phys.Chem.,1987, 38 ,463.
[3] S. Mabrey and J. M. Sturtevant, Proc. Natl. Acad., Sci. USA, 1976, 73 , 3862.
[4] From equation (b)
\[\begin{aligned}
&\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{\mathrm{d}}{\mathrm{dT}}\left[\mathrm{K}^{0} \,\left(1+\mathrm{K}^{0}\right)^{-1}\right]=\left[\frac{1}{1+\mathrm{K}^{0}}-\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}} \\
&=\frac{1}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\mathrm{dK}^{0}}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{1}{\mathrm{~K}^{0}} \, \frac{\mathrm{dK}^{0}}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{\mathrm{d} \ln \mathrm{K}^{0}}{\mathrm{dT}}\right]
\end{aligned} \nonumber \]
[5] \(\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \, \frac{1}{[\mathrm{~K}]^{2}} \, \frac{[1]}{[1]} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\frac{1}{[\mathrm{~K}]}\)
[6] \(\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]^{2}}{\left.\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \mathrm{K}\right]^{2}} \, \frac{[1]}{[1]^{2}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\)
[7] With \(\mathrm{h}=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\), the second term in equation (k) can be written as follows \(\mathrm{y}=\frac{\mathrm{h}^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}}\). The pattern formed by the dependence of \(\mathrm{y}\) on temperature is given by
\[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \,\left[\frac{1}{\left(1+\mathrm{K}^{0}\right)^{2}}-\frac{2 \, \mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{3}}\right] \frac{\mathrm{dK}^{0}}{\mathrm{dT}}-\frac{2 \, \mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{3} \,\left(1+\mathrm{K}^{0}\right)^{2}} \nonumber \]
Or,
\[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{2} \,\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dT}}-\frac{2}{\mathrm{~T}}\right] \nonumber \]
Since \(\mathrm{d} \ln \left(\mathrm{K}^{0}\right) / \mathrm{dT}=\mathrm{h} / \mathrm{R} \, \mathrm{T}^{2}\), then
\[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{3} \,\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{h}}{\mathrm{R} \, \mathrm{T}}-2\right] \nonumber \]
Hence the condition for an extremum in \(\mathrm{y}\) as a function of \(\mathrm{T}\) is \(\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{h}}{\mathrm{R} \, \mathrm{T}}-2=0\) Or \(\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)}=\frac{2 \, \mathrm{R} \, \mathrm{T}}{\mathrm{h}}\) Then
\[\mathrm{K}^{0}=\frac{1-(2 \, \mathrm{R} \, \mathrm{T} / \mathrm{h})}{1+(2 \, \mathrm{R} \, \mathrm{T} / \mathrm{h})} \nonumber \]
By definition \(h=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\). Therefore if the magnitude of \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\) is much larger than \(2 \, \mathrm{~R} \, \mathrm{T}\), the top of the bell shaped curve is reached at a temperature where \(\mathrm{K}^{0}\) is unity. In the general case, at approximately this temperature \(\mathrm{y}\) is a maximum.
[8] M. J. Blandamer, J. Burgess and J. M. W. Scott, Ann. Rep. Prog. Chem., Sect. C, Phys. Chem.,1985, 82 ,77.
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