1.5.3: Chemical Potentials- Solutions- General Properties
A key quantity in chemical thermodynamics is the chemical potential of chemical substance \(j\), \(\mu_{j}\). The latter is the differential dependence of Gibbs energy on amount of substance \(j\) at fixed \(\mathrm{T}\), \(\mathrm{p}\) and amounts of all other substances in the system [1].
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
An important point to note is that the conditions ‘fixed \(T\) and fixed \(p\)’ on the partial differential refer to intensive variables. These conditions are called Gibbsian in recognition of the development by Gibbs of the concept of thermodynamic potential for changes in the properties of a closed system at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\).
In general terms, the chemical potential of substance \(j\) is defined using analogous partial derivatives of the thermodynamic internal energy \(\mathrm{U}\), enthalpy \(\mathrm{H}\) and Helmholtz energy \(\mathrm{F}\).
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{F}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
With reference to a given closed system, thermodynamics defines macroscopic properties including volume \(\mathrm{V}\), Gibbs energy \(\mathrm{G}\), enthalpy \(\mathrm{H}\) and entropy \(\mathrm{S}\). Nevertheless we need to “tell” these thermodynamic variables that a given system probably comprises different chemical substances. The analysis is reasonably straightforward if we define the system under consideration by the ‘Gibbsian’ set of independent variables; i.e. \(\mathrm{T}\), \(\mathrm{p}\) and amounts of each chemical substance [2]. The analysis leads to the definition of a chemical potential for each substance \(j\), \(\mu_{j}\), in a closed system [3,4].
Footnotes
[1] It might be argued that we have switched our attention from closed to open systems because we are considering a change in Gibbs energy when we add \(\partial n_{j}\) moles of substance to the system. This comment is true in part. But what we actually envisage is something a little different. We take a closed system containing \(n_{1}\) and \(n_{j}\) moles of substances \(1\) and \(j\) respectively. We open the system, rapidly pop in \(\delta \mathrm{n}_{\mathrm{j}}\) moles of substance \(j\) and put the lid back on the system to return it to the closed state. Then the closed system contains \(\left(\mathrm{n}_{\mathrm{j}}+\delta \mathrm{n}_{\mathrm{j}}\right)\) moles of substance j so changes in chemical composition and molecular organisation follow producing a change in Gibbs energy at, say, fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\).
[2] G. N. Lewis, (with possibly one of the key papers in chemistry)
- Z.Phys.Chem.,1907, 61 ,129.
- Proc. Acad. Arts Sci.,1907, 43 ,259.
[Comment: Paper (a) is the German translation of paper (b).]
[3] The analysis presented here (a) is confined to bulk systems in the absence of magnetic and electric fields and (b) ignores surface effects.
[4] To quote E. Grunwald [J. Am. Chem. Soc., 1984, 106 , 5414] “any first derivative with respect to any variable of state at equilibrium is isodelphic”; see also E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997.