1.3.9: Calorimetry- Scanning
A given closed system is prepared using \(n_{1}\) moles of water (\(\lambda\)) and \(n_{\mathrm{X}}^0\) moles of solute \(\mathrm{X}\) at pressure \(p\) (\(\cong \mathrm{p}^{0}\), the standard pressure) and temperature \(\mathrm{T}\). The thermodynamic properties of the solution are ideal such that, at some low temperature, the enthalpy of the solution \(\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})\) is given by equation (a).
\[\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { low } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{X}}^{\infty}(\text { aq } ; \text { low } \mathrm{T}) \nonumber \]
The temperature of the solution is raised to high temperature such that the solution contains only solute \(\mathrm{Y}\), all solute \(\mathrm{X}\) having been converted to \(\mathrm{Y}\).
\[\mathrm{H}(\mathrm{aq} ; \text { high } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { high } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{Y}}^{\infty}(\text { aq; high } \mathrm{T}) \nonumber \]
At intermediate temperatures, a chemical equilibrium exists between solutes \(\mathrm{X}\) and \(\mathrm{Y}\). At temperature \(\mathrm{T}\), the chemical composition of the solution is characterised by extent of reaction \(\xi(\mathrm{T})\).
\[\begin{array}{llcc}
& \mathrm{X}(\mathrm{aq})<\overline{=} & \mathrm{Y}(\mathrm{aq}) & \\
\text { At low } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0} & 0 & \mathrm{~mol} \\
\text { At high } \mathrm{T} & 0 & \mathrm{n}_{\mathrm{x}}^{0} & \mathrm{~mol} \\
\text { At intermediate } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T}) & \xi^{\mathrm{eq}}(\mathrm{T}) & \mathrm{mol}
\end{array} \nonumber \]
For a solution where the thermodynamic properties are ideal, we define an equilibrium constant \(\mathrm{K}(\mathrm{T})\), at temperature \(\mathrm{T}\).
\[\mathrm{K}(\mathrm{T})=\xi^{\mathrm{eq}}(\mathrm{T}) /\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T})\right] \nonumber \]
By definition, the degree of reaction, \(\alpha(T)=\xi^{e q}(T) / n_{x}^{0}\).
\[\mathrm{K}(\mathrm{T})=\alpha(\mathrm{T}) \, \mathrm{n}_{\mathrm{X}}^{0} /\left[\mathrm{n}_{\mathrm{X}}^{0}-\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T})\right] \nonumber \]
\[\text { Therefore, } \quad \alpha(\mathrm{T})=\mathrm{K}(\mathrm{T}) /[1+\mathrm{K}(\mathrm{T})] \nonumber \]
At temperature \(\mathrm{T}\), the enthalpy of the aqueous solution is given by equation (f) where \(\mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})\) is the molar enthalpy of water(\(\lambda\)) in the aqueous solution again assuming that the thermodynamic properties of the solution are ideal.
\[\begin{array}{r}
\mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \,[1-\alpha(\mathrm{T})] \, \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \\
+\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T}) \, \mathrm{H}_{\mathrm{Y}}^{0}(\mathrm{aq} ; \mathrm{T})
\end{array} \nonumber \]
The limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})\) is given by equation (g).
\[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq} ; \mathrm{T})-\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \nonumber \]
From equation (f),
\[\begin{aligned}
\mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, & \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \\
&+\mathrm{n}_{\mathrm{x}}^{0} \, \alpha(\mathrm{T}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})
\end{aligned} \nonumber \]
We assume that \(\Delta_{r} H^{\infty}(\mathrm{aq})\) is independent of temperature. The differential of equation (h) with respect to temperature yields the isobaric heat capacity of the solution.
\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\lambda)+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{C}_{\mathrm{pX}}^{\infty}(\mathrm{aq}) +\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT} \nonumber \]
The term \((\mathrm{d} \alpha / \mathrm{dT})\) signals the contribution of the change of composition with temperature to the isobaric heat capacity of the system, the ‘relaxational’ isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\text {relax})\).
Thus
\[\mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT} \nonumber \]
\[\text { From equation (e)[1] } \frac{\mathrm{d} \alpha(\mathrm{T})}{\mathrm{dT}}=\frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}} \, \frac{\mathrm{d} \ln [\mathrm{K}(\mathrm{T})]}{\mathrm{dT}} \nonumber \]
But according to the van’t Hoff Equation,
\[\frac{\mathrm{d} \ln \mathrm{K}(\mathrm{T})}{\mathrm{dT}}=\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\mathrm{R} \, \mathrm{T}^{2}} \nonumber \]
\[\text { Hence [2], } \mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{x}}^{0} \, \frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}} \nonumber \]
The contribution to the molar isobaric heat capacity of the system from the change in composition of the solution is given by equation (n).
\[C_{p m}(\text { relax })=\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}} \nonumber \]
The dependence of \(C_{p m}(\text {relax})\) on temperature has the following characteristics.
- The plot forms a bell-shaped curve such that at very low and very high temperatures, \((C_{p m}(\text {relax})\) is zero [3-5].
- \((C_{p m}(\text {max})\) occurs at the temperature \(\mathrm{T}_{m}\).
- The area under the ‘bell’ equals the enthalpy of reaction [6,7]. A scanning calorimeter is designed to raise the temperature of a sample (solution) in a controlled fashion. The calorimeter uses controlled electrical heating while monitoring the temperatures of a sample cell and a reference cell containing just solvent. If the heat capacity of the cell containing the sample under investigation starts to increase the calorimeter records the fact that more electrical energy is required to raise the temperature of the sample cell by say \(0.1 \mathrm{~K}\) than required by the reference.
The tertiary structures of enzymes in aqueous solution are very sensitive to temperature. In the general case, an enzyme changes from, say, active to inactive form as the temperature is raised; i.e. the enzyme denatures. The change from active to inactive form is characterised by a ‘melting temperature’. The explanation is centred on the role of hydrophobic interactions in stabilising the structure of the active form. However the strength of hydrophobic bonding is very sensitive to temperature. Hence equation (n) forms the basis of an important application of modern differential scanning calorimeters [3] into structural reorganisation in biopolymers on changing the temperature [4-7]. The scans may also identify domains within a given biopolymer which undergo structural transitions at different temperatures [8].
Indeed there is evidence that a given enzyme is characterised by a temperature range within which the active form is stable. Outside this range, both at low and high temperatures the active form is not stable. In other words the structure of an enzyme may change to an inactive form on lowering the temperature [9,10]. The pattern can be understood in terms of the dependence of \(\left[\mu_{\mathrm{j}}^{0} / \mathrm{T}\right]\) on temperature where \(\left[\mu_{\mathrm{j}}^{0}\) is the reference chemical potentials for solute \(j\). In this case we consider the case where in turn solute j represents the active and inactive forms of the enzyme. There is a strong possibility that the plots of two dependences intersect at two temperatures. The active form is stable in the window between the two temperatures.
The analysis leading to equation (n) is readily extended to systems involving coupled equilibria [11,12]. The impact of changes in composition is also an important consideration in analysing the dependence on temperature of the properties of weak acids in solution [13- 15].
Footnotes
[1] \(\begin{aligned}
\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\left[\frac{1}{1+\mathrm{K}}\right.&\left.-\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1+\mathrm{K}-\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}} \\
&=\left[\frac{\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{d} \ln \mathrm{K}}{\mathrm{dT}}
\end{aligned}\)
[2] \(\mathrm{C}_{\mathrm{p}}(\text { relax })=[\mathrm{mol}] \, \frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]^{2}}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2}} \, \frac{[1]}{[1]}=\left[\mathrm{J} \mathrm{K}^{-1}\right]\)
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