Chemistry Homework
Q1. (a) Calculate the single-molecule translational partition function of I(g) [not I2 (g)] at 3000 K and 1 bar. For the volume, use the volume of 1 mol of an ideal gas under these conditions.
(b) Compare your result to the single-molecule translational partition function of H(g) under these same conditions. This value has already been calculated in Example 16.3.
(c) Explain in words why a heavier atom has a larger translational partition function than a lighter atom at the same temperature. An answer that Equation 16.39 has mass in the numerator is not sufficient.
Q2. (a) Table 16.1 gives the contributions to the molar thermodynamic properties of ideal gases under arbitrary conditions. Describe how to use this table to calculate the properties of ideal gases in the standard state.
(b) Calculate the molar entropy (in units of R) for He(g) in the standard state at 25oC. Assume that the ground state degeneracy of the He atom is 1.
(c) Compare your result to the experimental value given in Table C.2.
Q3. (a) Calculate the molar heat capacity at constant pressure (in units of R) for CO2 (g) in the standard state at 25oC. Note that the one of the characteristic vibrational temperatures occurs twice.
(b) Compare your result to the experimental value given in Table C.2.
Q4. Follow the steps below to derive the translational contribution to the molar entropy of an ideal gas. This will help you to understand how the expressions in Table 16.1 are derived.
(a) Start with the single-molecule partition function for translation (Equation 16.39). Use this to derive the canonical partition function for translation (Equation 16.41).
(b) Use your previous result and Equation 16.9 to derive the Helmholtz free energy (the final part of Equation 16.42). Hint: Do not forget about Stirling's approximation (Equation 16.20).
(c) Assuming that St = -(∂A/∂T)V,N, derive the translational contribution to entropy (Equation 16.43).
(d) Derive Equation 16.51 by converting your previous result to a function of T and P.
(e) Convert your previous (extrinsic) result to the molar entropy in units of R as given in Table 16.1.
Q5. On page 601, in the last paragraph before section 16.12, Silbey, Alberty, and Bawendi discuss the fact that Q is in error by a large amount under common conditions because of the multiple occupancy of levels. For example, at 25oC and 1 bar, a typical error is a factor of 10(-10^15). That's a big error!
(a) Calculate the magnitude of the change in the Helmholtz free energy of a system in joules once this error is included. To do this, determine ?A = A - A′, where is calculated using a canonical partition function Q and A′ is calculated with Q replaced by Q⋅10(-10^15). Hints: The answer is not zero. Your calculator will underflow if you try to calculate the answer directly, so use your brain first, the properties of logarithms second, and your calculator a distant third.
(b) Comment on whether this error is significant compared to the Helmholtz free energy change in a typical chemical reaction. Hint: ?rA is roughly the same order of magnitude as ?rG.
Textbook - Physical Chemistry, 4th Edition by Robert J. Silbey, Robert A. Alberty and Moungi G. Bawendi