Statistical physics (2005)

In the case of a monoatomic ideal gas, Cv only contains the contribution of

3

the three translation degrees of freedom and is thus equal to 2 N kB .

4.4.3Free Energy; Physical Parameters (P, S, µ)

We saw in chapter 3 that the canonical partition function Zc is related to the free energy through

In the case of an ideal gas, the di erential of F is

The volume Ω and the number of particles N are extensive variables, the temperature T is intensive. The chemical potential, already introduced in § 3.5.3, is here the variation of free energy of the system at fixed temperature and volume when one particle is added.

Substituting the expression of Zc obtained in §4.4.1, one obtains :

For the potential F to be extensive, the quantity between brackets must be

so that the constant in Eq. (4.39) is equal to unity. This choice will be justified in §4.4.4 by a physical argument, the Gibbs paradox, and will be demonstrated in §6.7. Then

One deduces the pressure, the entropy and the chemical potential of the assembly of N particles :

One identifies the first of these relations as the ideal gas law, that we now just re-demonstrated using Statistical Physics.

The expression obtained for S(Ω, N, T ) is extensive but does not satisfy the Nernst principle, since it does not vanish when the temperature tends to zero : the entropy would become very negative if the de Broglie thermal wavelength was becoming too large with respect to the mean distance between particles. This confirms what we already expressed in §4.3, that is, the ideal gas model is no longer valid at very low temperature.

In the validity limit of the ideal gas model, λth d is equivalent to Ω/N λ3th very large, thus to µ very negative (µ → −∞).

Remember that the measure of the classical N -particle phase space must be

mical potentials F and S. This will be justified in chapter 6 : the factor CN expresses the common classical limit of both Quantum Statistics.

4.4.4Gibbs Paradox

At the end of the 19th century, John Willard Gibbs used the following physical argument to deduce the requirement of the constant CN = N1! in the expression of the infinitesimal volume of the phase space : consider two containers of volumes Ω1 and Ω2, containing under the same density and at the same temperature T , respectively, N1 and N2 molecules of gas (Fig. 4.3). One then opens the partition which separates the two containers, the gases mix and the final total entropy is compared to the initial total one. If the gases in the two containers were of di erent chemical natures, the disorder and thus the entropy increase. On the other hand, if both gases are of the same nature and the molecules indistinguishable, the entropy is unchanged in the process.

The detailed calculation of the di erent terms in the total entropy, before

Fig. 4.3: Mixture of two gases at the same temperature and with the same density.

and after opening the partition, is simple and uses the partial derivative with respect of temperature of expression (4.38) of the free energy, including the constant CN : if one assumes that CN = 1, there will be a variation of the total entropy even for a mixture of molecules of the same nature. When one introduces the factor CN = 1/N !, the result corresponds to the physical prediction. The choice of this factor is not surprising once one considers that molecules of the same chemical nature are indistinguishable : for a system of N indistinguishable particles, there are N ! possible permutations of these particles among their one-particle states, providing the same macroscopic state. In order to account only once for this state in the enumeration that will lead to the physical predictions, among others of the entropy, it is natural to choose the value CN = 1/N !.

4.5Conclusion

Using Statistical Physics, we have been able to find again the well-known results of Thermodynamics on the ideal gas and to justify them. It seems interesting, now that we come to the end of the study of Classical Statistics, to present the respective contributions of Thermodynamics (Appendix 4.1) and of Statistical Physics (Appendix 4.2) on a practical example. This will be done on the analysis of the chemical reactions.

In the Statistical Physics part, one will show in details the various microscopic terms of the hamiltonian of a gas of diatomic molecules; this allows one to

understand the origin of a law very commonly used in Chemistry, that is, the law of mass action. Enjoy your reading!

Reading these two Appendixes is not a prerequisite to understanding the next chapters.

Chemical reactions are mostly done at constant temperature : one then has to consider the case of a system A in thermal contact with a heat reservoir A at temperature T0 (Fig. 4.4). The combined system Atot, made of A and A is an isolated system, the entropy Stot of which increases in a spontaneous process :

The entropy variation of the combined system

can be expressed as a function of parameters of the system A alone.

Indeed, if the amount of heat Q is given to A by the heat reservoir, the reservoir absorbs −Q and its entropy variation is equal to ∆S = −Q/T0 : the transfer of Q does not produce a variation of T0 and one can imagine a reversible sequence of isothermal transformations at T0 producing this heat exchange.

Besides, the internal energy variation in A, i.e., ∆U , is, from the first law of Thermodynamics,

where W is the work operated in the considered transformation. Consequently,

where, by definition, F0 = U − T0S, which is identical to the Helmholtz free energy F = U − T S of system A if the temperature of A is that of the heat reservoir (in an o -equilibrium situation T di ers from T0 and F is not identical to F ).

From the condition of spontaneous evolution one deduces that

which expresses that the maximum work that may be delivered by A in a reversible process is (−∆F0), whence the name of “free energy.”

If the volume of system A is maintained constant and there is no other type of work, W = 0, so that the condition for spontaneous evolution is then given by

This condition for a system in contact with a heat reservoir replaces the condition ∆Stot ≥ 0 for an isolated system.

A more realistic situation is the one in which system A is kept at constant temperature and constant pressure. This is the case in laboratory experiments when a chemical reaction is performed at fixed temperature T0 and under the atmospheric pressure P0 : system A is exchanging heat and work with a very large reservoir A , which can thus give energy to A without any modification of T0 and also exchange a volume ∆Ω with A without any change of its pressure P0 (Fig. 4.5).

Atot

Fig. 4.5: System A is in contact with the larger system A , with which it is exchanging the amount of heat Q and the volume ∆Ω.

Similarly, one considers that the combined system made of A and A is isolated, so that in a spontaneous process

If A absorbs Q from A , ∆S = −Q/T0, where T0 is again the temperature of

A .

The variation of internal energy of A includes the received heat Q, the mechanical work −P0∆Ω received by A from A , and also W , work of any other origin (electrical for example) that A may have received :

Here we defined G0 = U + P0Ω − T0S, which identifies to the Gibbs free enthalpy of system A, that is, G = U + P Ω − T S, when P = P0 and T = T0.

A spontaneous evolution will be expressed through the condition

When all the external parameters of system A, except its volume, are kept fixed, W is null and the condition of spontaneous evolution is then

Consequently, when a system is in contact with a reservoir so that it remains at fixed temperature and pressure and can exchange work only with this reservoir, the stable equilibrium is characterized by the condition of minimum of G0.

2. Chemical Equilibrium, Law of Mass Action : Thermodynamical

Approach

Consider a homogeneous system (for example in gaseous phase) with several types of molecules : here we will study the dissociation reaction of iodine vapor at temperatures in the 1000 K range :

I2 2I

In this reaction the number of iodine atoms is conserved between the species corresponding to the two members of the chemical equation, which can also be expressed in the form

2I − I2 = 0

the species disappearing in the reaction carrying a negative sign, the one appearing a positive one. In the same way, the most general chemical equilibrium will be written

n

i=1

The bi are the integral coe cients of the balanced reaction for the Bi molecules. The number Ni of molecules of Bi type in the system is changed if the equilibrium is displaced but the total number of atoms of each species in the system is conserved. The variations of the numbers Ni are thus proportional to the coe cients bi of the chemical equilibrium

dNi = bi dλ for any i

where dλ is a proportionality constant. dNi is positive for the molecules appearing, negative for those vanishing in the reaction.

Consider a chemical equilibrium at fixed temperature T and pressure P , thus characterized by dG(T, P ) = 0. In such conditions the equilibrium is expressed by :

by its expression versus λ, one obtains the relation characterizing the equilibrium :

n

i=1

The chemical potentials depend on the experimental conditions, in particular on the concentrations of the di erent species. Assume that these reagents are ideal gases and look for the dependence of µi on the partial pressure Pi of one of these gases.

Since Gi is extensive and depends on a single extensive parameter Ni, the number of molecules of the species i, then one must have

The dependence in temperature and pressure of µi is thus that of Gi. For an ideal gas, at fixed temperature,

introducing the partial pressure Pi of species i, such that PiΩ = NikB T . Consequently,