1.3.3: Calorimetry- Solutions- Isobaric
Classic (isobaric) calorimetric experiments often centre on the determination of the change in enthalpy ∆H for a given well-defined process. For example, the heat accompanying the mixing of known amounts of two liquids [e.g. water(\(\lambda\)) and ethanol(\(\lambda\))] to form a binary liquid mixture yields the enthalpy of mixing, \(\Delta_{\mathrm{mix}}\mathrm{H}\). Similarly enthalpies of solution are obtained by recording the heat accompanying the solution of a known amount of solute (e.g. urea) in a known amount of solvent; e.g. water(\(\lambda\)). Key equations emerge from the following analysis.
The enthalpy \(\mathrm{H}\) of a closed system is an extensive function of state which for a closed system is defined by the set of independent variables, \(\mathrm{T}\), \(p\) and \(\xi\) where \(\xi\) represents the chemical composition.
\[\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi] \label{a} \]
Equation \ref{b} is the complete differential of Equation \ref{a}.
\[\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{b} \]
If the closed system is held at constant pressure (e.g. ambient) the differential enthalpy \(\mathrm{dH}\) equals the heat \(\mathrm{dq}\).
\[T\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \nonumber \]
Here \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\), is the differential dependence of enthalpy \(\mathrm{H}\) on temperature at constant pressure and composition whereas \(\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\), is the differential dependence of enthalpy \(\mathrm{H}\) on composition at fixed temperature and pressure.