1.8.2: Enthalpy
- Page ID
- 374765
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There is considerable merit in identifying an extensive property of a closed system called the enthalpy, \(\mathrm{H}\). The enthalpy of a closed system is a state variable and defined by equation (a).
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
We identify a given state by the symbol I having enthalpy \(\mathrm{H}[\mathrm{I}]\), energy \(\mathrm{U}[\mathrm{I}]\) and volume \(\mathrm{V}[\mathrm{I}]\) at pressure \(\mathrm{p}\).
\[\mathrm{H}[\mathrm{I}]=\mathrm{U}[\mathrm{I}]+\mathrm{p} \, \mathrm{V}[\mathrm{I}] \nonumber \]
This system is displaced to a neighbouring state such that the differential change in enthalpy is \(\mathrm{dH}\). Using equation (a),
\[\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
But according to the first law of thermodynamics, the differential change in thermodynamic energy \(\mathrm{dU}\) is given by ‘\(q-p \, d V\)’ where \(\mathrm{q}\) is the heat accompanying the change. Then,
\[\mathrm{dH}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
or,
\[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp} \nonumber \]
At constant pressure,
\[\mathrm{dH}=\mathrm{q} \nonumber \]
For a change from state I to state II the change in enthalpy is given by equation (g).
\[\Delta \mathrm{H}=\int_{\mathrm{I}}^{\mathrm{II}} \mathrm{dH}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q} \nonumber \]
In equation (g) we replace the integral of dH by the difference \(\mathrm{H}(\mathrm{II}) - \mathrm{H}(\mathrm{I})\) because enthalpy is a state variable and so \(\Delta \mathrm{H}\) is independent of the path between the two states and hence so is \(\mathrm{q}\). In liquid solutions, the recorded heat is also independent of the rate of change in chemical composition between state I and state II.