1.14.22: Descriptions of Systems
- Page ID
- 374483
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Important themes in thermodynamics involve (i) properties (variables) which can be measured (e.g. volumes and/or densities) and (ii) thermodynamic variables which are rigorously defined (e.g. enthalpies). In these terms a measured property (e.g. density) is the reporter of the chemical properties or processes taking place in a system. So it is always important to ask if the “reporter” can be interrogated for the required information. In fact there is often a limit to the amount of information which a given reporter offers to the investigator. These important limitations should be borne in mind. An example makes the point.
A system is prepared by placing \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of substance in a closed vessel at fixed \(\mathrm{T}\) and \(\mathrm{p}\). [The superscript ‘0’ implies at zero time.] We explore two possible descriptions of this system. Perhaps two samples were analysed by two independent laboratories.
Description A
The first laboratory reports that the system is simple and contains the single substance \(\mathrm{X}\).
Gibbs energy
\[\mathrm{G}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mu_{\mathrm{X}}^{*}(\ell) \nonumber \]
and volume
\[\mathrm{V}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{X}}^{*}(\ell) \nonumber \]
Here \(\mu_{X}^{*}(\ell)\) and \(\mathrm{V}_{\mathrm{x}}^{*}(\ell)\) are the chemical potential and molar volume of the pure chemical substance \(\mathrm{X}\).
Description B
The second laboratory identifies two substances \(\mathrm{X}\) and \(\mathrm{Y}\) in chemical equilibrium such that the equilibrium amounts of substances \(\mathrm{X}\) and \(\mathrm{Y}\) are respectively \(\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}}\).
Gibbs energy
\[\mathrm{G}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mu_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mu_{\mathrm{Y}}^{\mathrm{eq}} \nonumber \]
and volume
\[\mathrm{V}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{Y}}^{\mathrm{eq}} \nonumber \]
Here \(\mu_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mu_{\mathrm{Y}}^{\mathrm{eq}}\) are the equilibrium chemical potentials; \(\mathrm{V}_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mathrm{V}_{\mathrm{Y}}^{\mathrm{eq}}\) are the equilibrium partial molar volumes.
Description A is “primitive” and Description B is “sophisticated”. Both Gibbs energies and volumes are functions of state so that \(\mathrm{V}(\mathrm{A})=\mathrm{V}(\mathrm{B})\) and \(\mathrm{G}(\mathrm{A})=\mathrm{G}(\mathrm{B})\). The chemical potential of substance \(\mathrm{X}\) describes the change in \(\mathrm{G}\) when \(\delta n_{X}\) moles of \(\mathrm{X}\) are added. This chemical potential is insensitive to the changes taking place in the equilibrium system;\(\mu_{X}(\mathrm{~A})=\mu_{X}(\mathrm{~B})\). Consequently, measurement of volume \(\mathrm{V}\) [and if it were possible of \(\mathrm{G}\)] would not distinguish between the two descriptions. Similarly, measurement of \(\mathrm{H}\) (if it were possible) would not distinguish between the two descriptions.
Footnotes
[1] L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd edn., 1970,p.16.