Chemistry 252 Problem Set 2
1. Calculation of average energies from partition functions Consider a system that has only three possible energy states with E = 0, E0 and 2E0.
a) Write a general expression for the partition function q.
b) From your expression in (a), derive an expression for the average energy E as a function of temperature T.
c) Derive an expression for the specific heat Cv.
d) Suppose that E0 = 100 cm-‐1. Estimate the energy needed to raise the
temperature of the system from 300 K to 400 K in units of KJ/mol. 2. Specific heats of real gaseous molecules A review of sections 4-‐6 and 4-‐7 of the textbook will be useful in solving this problem.
a) Study equation 4.39 of the text and the resulting partition function derived for a gaseous diatomic molecule in example 4-‐5 of the text. Use these expressions for Q(N,V,T) to derive an expression for the specific heat.
b) Interpret each of the terms in the expression you derived. This is actually
done in the text (and we reviewed it in class) so it should be easy but I thought it was so important that I’d like you to put what they say into your own words.
c) Now, I’d like you to generalize the expression for the specific heat Cv to be
appropriate to a polyatomic molecule, specifically H2O(g). Assume that we’ll be working in the temperature range from 300 – 800 K where we can consider ourselves to be way above the rotational temperature Θrot so you can just approximate the rotational contribution to specific heat to be 3R/2 per mole corresponding to R/2 for each of the three rotational degrees of freedom. The main modification you need to make is to consider that there are 4 degrees of vibrational freedom for water (asymmetric stretch ħω1 = 3756 cm-‐1, symmetric stretch ħω2 = 3652 cm-‐1, bending mode ħω3 = 1595 cm-‐1 and another bending mode ħω4 = 1595 cm-‐1).
d) Use your expression from c) to plot Cv versus temperature over the
temperature range specified above (calculating a point every 100 K should be enough). Compare your result to experimental values from the literature.
e) As is evident from Figure 4.7, there is excellent agreement of calculated and
measured specific heats. The text notes (page 160) that the agreement can be improved still further if we refine the harmonic oscillator model to consider anharmonicity – i.e. the fact that the potential is not really harmonic. (For reference, see problem 1-‐27 and 1-‐31 of the text on page 34). Given that accounting for anharmonicity decreases the spacings between energy levels relative to what they would have been in a completely harmonic system, reason as to whether making a correction for anharmonicity would increase or decrease the values of your calculated specific heat. Explain your reasoning.