William M. White - Geochemistry

(2.40)

and

(2.41)

As additive properties, ln and ln Ω are consistent with our other extensive state variables (e.g., U , V ).

We want to know which energy distribution (i.e., values of e and E – e ) is the most likely one, because that corresponds to the equilibrium state of the system. This is the same as asking where the probability function, ( e ), is maximum. Maximum values of functions have the useful property that they occur at points where the derivative of the function is 0 . That is, a maximum of function ƒ(x) will occur where d ƒ(x) /d (x) = 0. *Thus, the maximum value of in Figure 2.8occurs where d /d e = 0. The most probable energy distribution will therefore occur at:

(2.42)

(we use the partial differential notation to indicate that, since the system is isolated, all other state variables are held constant). Substituting eqn. 2.41into 2.42, we have:

(2.43)

(since C is a constant). Then substituting eqn. 2.40into 2.43we have:

(2.44)

so the maximum occurs at:

(2.45)

The maximum then occurs where the function ∂lnΩ/∂e for each of the two blocks are equal (the negative sign will cancel because we are taking the derivative ∂ƒ(−e)/∂e). More generally, we may write:

(2.46)

Notice two interesting things: the equilibrium energy distribution is the one where ln Ω is maximum (since it is proportional to P) and where the functions ∂lnΩ/∂E of the two blocks are equal . It would appear that both are very useful functions. We define entropy, S , †as:

(2.47) †

and a function β such that:

(2.48)

where k is a constant (which turns out to be Boltzmann's constant or the gas constant; the choice depends on whether we work in units of atoms or moles, respectively). The function S then has the property that it is maximum at equilibrium and β has the property that it is the same in every part of the system at equilibrium .

Entropy also has the interesting property that in any spontaneous reaction, the total entropy of the system plus its surroundings must increase. In our example, this is a simple consequence of the observation that the final probability, ( E ), and therefore also Ω, will be maximum and hence never be less than the original one. Because of that, the final number of accessible states must exceed the initial number and:

(2.49)

rearranging:

The quantities in brackets are simply the entropy changes of the two blocks. Hence:

(2.50)

In other words, any decrease in entropy in one of the blocks must be at least compensated for by an increase in entropy of the other block.

For an irreversible process, that is, a spontaneous one such as thermal equilibrium between two copper blocks, we cannot determine exactly the increase in entropy. Experience has shown, however, that the increase in entropy will always exceed the ratio of heat exchanged to temperature. Thus, the mathematical formulation of the second law is:

(2.51)

Like the first law, eqn. 2.51cannot be derived or formally proven; it is simply a postulate that has never been contradicted by experience. For a reversible reaction, that is, one that is never far from equilibrium and therefore one where d Q is small relative to T ,

(2.52)

(see Example 2.1). In thermodynamics, we restrict our attention to systems that are close to equilibrium, so eqn. 2.52serves as an operational definition of entropy.

Let's now return to our function β . The macroscopic function having the property of our new function β is temperature. The relation of temperature to β is

(2.53)

and

(2.54)

equation 2.53provides a statistical mechanical definition of temperature. We can easily show that T is the measure of the energy per degree of freedom . To do this, though, we need one other relationship, which we introduce without proof. This is that Ω increases roughly with E as:

where ƒ is the number of degrees of freedom of the system (which, in turn, is proportional to the number of atoms or molecules in the system times the modes of motion, e.g., vibrational, rotational, translational, available to them). Hence: