1.14.41: Lewisian Variables
- Page ID
- 389718
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. If the thermodynamic properties of the liquid mixture are ideal the volume of the mixture is given by the sum of products of amounts and molar volumes (at the same \(\mathrm{T}\) and \(\mathrm{p}\)); equation (a).
\[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
If the thermodynamic properties of the mixture are not ideal, the volume of the (real) mixture is given by equation (b).
\[V(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
\(\mathrm{V}_{1}(\operatorname{mix})\) and \(\mathrm{V}_{2}(\operatorname{mix})\) are the partial molar volumes of chemical substances 1 and 2 defined by equations (c) and (d).
\[\mathrm{V}_{1}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{l}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(2)} \nonumber \]
\[\mathrm{V}_{2}(\operatorname{mix})=\left(\frac{\partial V}{\partial n_{2}}\right)_{T_{, p, n(1)}} \nonumber \]
The similarities between equations (a) and (b) are obvious and indicate an important method for describing the extensive properties of a given system. This was the aim of G. N. Lewis who sought equations of the form show in equation (b). In general terms we identify an extensive property \(\mathrm{X}\) of a given system such that the variable can be written in the general form shown in equation (e).
\[\mathrm{X}=\mathrm{n}_{1} \, \mathrm{X}_{1}+\mathrm{n}_{2} \, \mathrm{X}_{2} \nonumber \]
where
\[\mathrm{X}_{1}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(2)} \nonumber \]
\[\mathrm{X}_{2}(\operatorname{mix})=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
Other than the composition variables, the conditions on the partial differentials in equations (f) and (g) are intensive properties;
- mechanical variable, pressure, and
- thermal variable, temperature.
Lewisian partial molar variables can be used to describe the thermodynamic energy \(\mathrm{U}\), entropy \(\mathrm{S}\) and volume \(\mathrm{V}\) together with their Legendre transforms, Helmholtz energy, enthalpy and Gibbs energy. With respect to other thermodynamic properties of a closed system, the case for identifying similar Lewisian partial molar properties has to be established. It turns out that partial molar expansions [e.g. \(\mathrm{E}_{\mathrm{p} j}(\mathrm{T}, \mathrm{p})\)] and partial molar compressions [e.g. \(\mathrm{K}_{\mathrm{T} j}(\mathrm{T}, \mathrm{p})\)] for chemical substance \(j\) in a closed single phase system are Lewisian but partial molar isentropic compressions are not .