Examples from the monatomic ideal gas case

In problem 3.2 you showed that example 1 1

Computing C v for Argon including first 4 excited states State  eV ii i=101 i= i= i= i=

Example 2. Calculate the numerical value of the canonical partition function for a single He atom (q) in a cubic box of edge 1 cm and the probability of finding the He atom in a single energy level corresponding to the mean kinetic energy of the molecule at 300K a) The individual partition function for He is we can set  n1 = 1 and  e1 = 1 since there is no degeneracy for He (all the electrons fill the 1s orbital such that the spin of the electrons are nondegenerate) Let M = m ∙ 6.02x

b) the probability of finding the He atom in a single energy level corresponding to the mean kinetic energy of the molecule at 300K c) The probability of finding He within ±1% of the mean energy: 4

Example 3. Consider the mixing of N A molecules of monatomic gas A and N B molecules of monatomic gas B at constant volume V and temperature T. Write the partition function for this system in terms of q A and q B. Develop expressions for E, C v, P, and the entropy S=S(x A, x B, P, T); x A and x B are mole fractions. Also calculate the entropy of mixing. with 5

6

entropy of mixing 7

Example 4. An ideal gas of monatomic molecules on a line (for example in a nanopore) where they cannot pass each other  the translational motion is restricted to one dimension. The qm translational energy levels are: L x is the length of the line (pore). Find the translational contribution to the partition function and the thermodynamic properties of this gas. 8 Replacing a summation with an integral, we immediately obtain:

Example 5. A large polymer molecule is made up to N monomer units, each of which can be either a helix (H) or a coiled (C) state with energies  H and  C respectively. Assuming that the conformation of a monomer unit is independent of all other monomer units, determine the average fraction of monomers that are in the H state as a function of a dimensionless T. How does the statistical degeneracy comes into the result? 9

Example 5. A large polymer molecule is made up to N monomer units, each of which can be either a helix (H) or a coiled (C) state with energies  H and  C respectively. Assuming that the conformation of a monomer unit is independent of all other monomer units, determine the average fraction of monomers that are in the H state as a function of a dimensionless T. How does the statistical degeneracy come into the result? 10 The total energy of the polymer chain (N monomers) depends on the number of monomers in the helix state (n) and coiled state (N-n), i.e., The number of combinations of n helix monomers and N-n coiled monomers is Therefore, the partition function of a polymer chain is the last equality is obtain from binomial expansion, see eqn in the textbook

11 The probability of finding the polymer having n helix monomers is The average number of helix monomers is:

12 Similarly the average number of coiled monomers is The average number of monomers that are in the helix state can be rewritten as: with

Example 6. Compute the chemical potential of Avogadro’s number of molecules of Ar at 298K and 1 bar pressure 13 First need to calculate the volume of 1 mole of an ideal gas at 298 K and 1 bar