In all the thermodynamic functions we have studied so far we have ignored the fact that the chemical composition of the system might change. In fact, the functions of state might be summarized by:
where v1,..., vK indicate the number of moles of each chemical species in the system. These molar indicators are thermodynamic coordinates. We can write
We can write more compactly dG = Vdp - SdT + G1 dv1 + ••• + Gn dvn (4.116)
where the Gi are the molar Gibbs energies for the individual components in the mixture.1 Note that a similar expression (but with differentials of their natural variables serving as coefficients) holds for dU and dH with the same values of Gi.
H = H (S, p, V1, ..., Vn) G = G(T, p, V1, ..., Vn) S = S (U, V, V1, ..., Vn)
expresses how much the composite Gibbs energy changes per mole of species 1 being added to the system. As with other intensive parameters in subsystems in contact (such as T1 and T2 when the subsystems 1 and 2 are in diathermal contact, or p1 and p2 when the pressures are allowed to be unconstrained) the specific Gibbs energies Gi will tend toward equality when the number of moles of the different species are allowed to vary (e.g., by chemical reactions or phase changes).
We can obtain some insight into this equalizing of the Gi by considering a system at constant pressure and temperature in which there are two chemical species, A and B. We have the reaction:
1 The Gi are denoted /¿i in the chemical literature and are called the chemical potentials of the system components.
The number of moles of B being created, Svb = — Sva. And since
t— ) dvA + — dvB = (Ga — Gb) dvA. (4.119) dvA J t p V9vb/ T ,p
The species A and B might be different phases of the same substance. We again find the equality of the two molar Gibbs energies for each phase when equilibrium is established. We will find this to be of great utility in the next chapter.
Next consider a system composed of two subsystems of equal volume, one is filled with vA moles of ideal gas species A the other with vB moles of B. Further, suppose the two gases have the same pressure p and temperature T. Now suppose the two subsystems are brought into material contact with the volume being the sum of the original volumes, the pressures and temperatures also being the same. What is the final Gibbs energy? What are the final enthalpy, internal energy, and entropy?
The initial Gibbs energy is:
where we have used the same specific Gibbs energy G(p, T) for each of the ideal gases A and B. Once the gases are mixed into the larger volume, the total pressure will be the same, but the partial pressures will be only half as much since they occupy twice the volume but at the same temperature. Hence,
Taking the difference and using the formula (4.101) we get
AG = —final — Ginit = — (va + vb)R*T ln 2 < 0. (4.123)
This illustrates that the spontaneous process of mixing two ideal gases leads to a decrease in the Gibbs energy.
For the internal energy and enthalpy, the job is easy. The change of the internal energy is
since AT = 0. The same holds for enthalpy with the substitution cV ^ cp.
As expected, during the mixing of ideal gases the Gibbs energy decreases, while the enthalpy and internal energies do not change. To calculate the entropy change we choose a reversible isothermal path. We use
The mixing of two ideal gases causes an increase of the entropy as we learned earlier.
Suppose now that the gases A and B are identical, MA = MB = M, RA = Rb = R, vA = vB = v. Then, from (4.126) we get the increase of the entropy after mixing:
Does this make sense? In the beginning each subvolume contains the same number of moles of identical gases. What changes after the mixing of the gases? Nothing. Then the change in entropy should be zero. So we get two different answers for the same problem. This has become known as the Gibbs Paradox. The reason this paradox arises is that in classical physics we cannot consider the mixing of two identical gases as a limiting case of the mixing of two different gases. If we start our consideration for different gases, they have always to be different. It is impossible to get the answer for the entropy change of the mixing of two identical gases simply by equating the masses and the gas constants in equation (4.126). In classical physics the exchange of coordinates between two identical particles (gas molecules in our case) corresponds to a new microscopic state of the system (two gases in the cylinder), although nothing changes with such an exchange at the macroscopic level. This paradox does not exist in quantum theory, where the exchange of two identical particles does not correspond to a new microscopic state of the system. Therefore, when two identical gases are mixed, the entropy does not change.
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