1.3.3: Calorimetry- Solutions- Isobaric
- Page ID
- 352515
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.