1.14.22: Descriptions of Systems

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

\[\mathrm{G}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mu_{\mathrm{X}}^{*}(\ell)\]

and volume

\[\mathrm{V}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{X}}^{*}(\ell)\]

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

\[\mathrm{G}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mu_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mu_{\mathrm{Y}}^{\mathrm{eq}}\]

and volume

\[\mathrm{V}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{Y}}^{\mathrm{eq}}\]

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.