Abstract
In Chap. 5, we noted that in combination the first and the second laws lead to an important relationship: the difference, dU, in the internal energy of two neighboring equilibrium states is linearly related to the corresponding difference, dS, in their entropy. Of course, also included in this relationship is the heat energy quasi-statically added to the system and the quasi-static work, \(\mathrm{d}{W}_{\mathrm{quasi-static}} = P\mathrm{d}V\), performed by the system when it transitions from a state with extensive variables (U, S, V ) to one with \((U + \mathrm{d}U,S + \mathrm{d}S,V + \mathrm{d}V )\).
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Notes
- 1.
And, of course, the enthalpy H – which is an extensive variable – that has also been considered before. Here the inclusion of H, however, is subsumed in that of U because the knowledge of the volume (and its conjugate variable, the pressure) relates U to H.
- 2.
Although, in the chapter on imperfect gases – see Chap. 6 – we did describe the co-existence of liquid–vapor phases, it was done without an explicit treatment of the internal energy, U, and the entropy S, in the co-existent regime. Therefore, we were able to make do without having to introduce the concept of varying n.
- 3.
Note: According to (8.3), the rate of change of the extensive function U is completely described in terms of the rates of change of the extensive variables S, V, n j ’s and \({\mathcal{X}}_{i}\)’s. Consequently, U is a function only of these extensive quantities. Moreover, because dU is an exact differential, its partial differentials with respect to any of the extensive variables are equal to, what we shall call, their conjugate intensive fields. For instance:
$$\left (\frac{\partial U} {\partial S} \right ) V,\,{n}_{j},{\mathcal{X}}_{i} = T;\ \left (\frac{\partial U} {\partial V }\right ) S,{n}_{j},{\mathcal{X}}_{i} = -P;\ \ \ \left ( \frac{\partial U} {\partial {n}_{l}}\right ) S,{n}_{j\neq l},{\mathcal{X}}_{i} = {\mu }_{l};\ \mathrm{etc}.$$(8.4)The subscripts in the above equation that include n j need to be summed over all values of j. For notational convenience, the sum is not displayed.
- 4.
Usually, λ is called the “scaling parameter.”
- 5.
Note: Here λ is the parameter.
- 6.
Occasionally, the Euler equation is also called the “Complete Fundamental Equation.”
- 7.
op. cit.
- 8.
For instance, multiply S, V, n j and \({\mathcal{X}}_{i}\) each by λ and U changes to λ U.
- 9.
Note that the phrase “energy representation” implies that the derivatives being considered here are those of the internal energy U.
- 10.
Note that the phrase “entropy representation” implies that the derivatives being considered here are those of the system entropy S.
- 11.
See, e.g., the chapter on the Ideal Gas.
- 12.
The possible three equations of state in the entropy representation are specified in the first three equations in (8.19).
- 13.
Clearly, for the more general system the Gibbs–Duhem equation in the energy representation would be
$$0 = S\mathrm{d}T - V \mathrm{d}P +{ \sum \nolimits }_{j}{n}_{j}\mathrm{d}{\mu }_{j} -{\sum \nolimits }_{i}{\mathcal{X}}_{i}\mathrm{d}{\mathcal{Y}}_{i}.$$ - 14.
For the more general system, in the entropy representation, the Gibbs–Duhem equation would be
$$0 = Ud\left ( \frac{1} {T}\right ) + V d\left (\frac{P} {T}\right ) -{\sum \nolimits }_{j}{n}_{j}d\left (\frac{{\mu }_{j}} {T} \right ) +{ \sum \nolimits }_{i}{\mathcal{X}}_{i}d\left (\frac{{\mathcal{Y}}_{i}} {T} \right ).$$ - 15.
For instance, look at (8.21) and note how it involves μ, T, S, n, U, and V.
- 16.
Remember, here we are considering the case where, in (8.25), \(\mathcal{X}\left (\frac{\mathcal{Y}} {T}\right ) = 0.\)
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Tahir-Kheli, R. (2012). Fundamental Equation and the Equations of State. In: General and Statistical Thermodynamics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21481-3_8
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DOI: https://doi.org/10.1007/978-3-642-21481-3_8
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