Examples other partition functions

Example 1. Show that For a grand canonical ensemble but we don’t know which thermodynamic property is related wi th. 2

In order to find that, we say that there is a function, f, which is function only of and represents a macroscopic thermodynamic function. That is from classical thermodynamics: and then therefore (1) (2) From (1) and (2): f = PV 3

we showed that: but also: 4

Example 2. A flat graphite surface is in contact with a reservoir of gas molecules at a fixed chemical potential. There are M sites on the graphite surface on which gas molecules can adsorb, each site can adsorb only a single molecule, and the adsorbed molecules do not interact with each other. Starting from the grand canonical partition function, develop an expression for the coverage—that is, the fraction of the adsorption sites that are occupied (contain an adsorbed gas molecule) as a function of the chemical potential of the gas molecules in the reservoir. Hints: 1.The number of ways of distributing N indistinguishable molecules on M adsorption sites that are distinguishable because they are fixed in space is 2. Using the binomial expression (appendix of Chapter 5) with q AD being the partition function for a site containing a molecule A adsorbed, and q B =1 for an empty site. 5

6 Here the term (1) M-N has been introduced in order to use the binomial expansion in the Appendix to Chapter 5. Therefore

7 then:  Langmuir isotherm??? To proceed, we recognize that equilibrium the chemical potential of the adsorbed molecules must be the same as that of the gas with which they are in contact.

8 The chemical potential of the gas molecules is: Langmuir isotherm check also the example in section 5.6