Chapter 11

The statistical mechanics of oxygen gas. Consider a system of one mole of ${\mathrm{O}}_2$ molecules in the gas phase at $T=273.15\mathrm{K}$ in a volume $V=22.4\times {10}^{-3}{\mathrm{m}}^3$. The molecular weight of oxygen is 32 . (a) Calculate the translational partition function $q_{\text{translation- }}$ (b) What is the translational component of the internal energy per mole? (c) Calculate the constant-volume heat capacity.

The statistical mechanics of a basketball. Consider a basketball of mass $m=1\mathrm{kg}$ in a basketball hoop. To simplify, suppose the hoop is a cubic box of volume $V=1{\mathrm{m}}^3$. (a) Calculate the lowest two energy states using the particle-in-a-box approach. (b) Calculate the partition function at $T=300\mathrm{K}$. Show whether quantum effects are important or not. (Assume that they are important only if $q$ is smaller than about $10.$ )

The translational partition function in two dimensions. When molecules adsorb on a two-dimensional surface, they have one less degree of freedom than in three dimensions. Write the two-dimensional translational partition function for an otherwise structureless particle.

The accessibility of rotational degrees of freedom. Diatomic ideal gases at $T=300\mathrm{K}$ have rotational partition functions of approximately $q=200$. At what temperature would $q$ become small (say $q<10$ ) so that quantum effects become important?

The entropy of crystalline carbon monoxide at $T=0\mathrm{K}.$ Carbon monoxide doesn't obey the 'Third Law of Thermodynamics': that is, its entropy is not zero when the temperature is zero. This is because molecules can pack in either the $\mathrm{C}=\mathrm{O}$ or $\mathrm{O}=\mathrm{C}$ direction in the crystalline state. For example, one packing arrangement of 12 CO molecules could be: $\begin{array}{llll}C=O& C=O& C=O& C=O\\ C=O& C=O& O=C& O=C\\ O=C& O=C& C=O& C=O\end{array}$ Calculate the partition function and the entropy of a carbon monoxide crystal per mole at $T=0\mathrm{K}$.

Temperature-dependent quantities in statistical thermodynamics. Which quantities depend on temperature? (a) Planck's constant $h$. (b) Partition function $q$. (c) Energy levels ${\epsilon }_j$. (d) Average energy $⟨\epsilon ⟩$. (e) Heat capacity $C_V$ for an ideal gas.

Heat capacities of liquids. (a) $C_V$ for liquid argon (at $T=100\mathrm{K}$ ) is $18.7{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$. How much of this heat capacity can you rationalize on the basis of your knowledge of gases? (b) $C_V$ for liquid water at $T={10}^{\circ }\mathrm{C}$ is about $75{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$. Assuming water has three vibrations, how much of this heat capacity can you rationalize on the basis of gases? What is responsible for the rest?

The entropies of CO. (a) Calculate the translational entropy for carbon monoxide $\mathrm{CO}$ ( $\mathrm{C}$ has mass $m=12$ amu, $\mathrm{O}$ has mass $m=16\mathrm{amu}$ ) at $T=300\mathrm{K},p=1\mathrm{atm}$. (b) Calculate the rotational entropy for $\mathrm{CO}$ at $T=300\mathrm{K}$. The CO bond has length $R=1.128\times {10}^{-10}\mathrm{m}$.

Conjugated polymers: why the absorption wavelength increases with chain length. Polyenes are linear double-bonded polymer molecules $(\mathrm{C}=\mathrm{C}-\mathrm{C}{)}_N$, where $N$ is the number of $\mathrm{C}=\mathrm{C}-\mathrm{C}$ monomers. Model a polyene chain as a box in which $\pi$-electrons are particles that can move freely. If there are $2N$ carbon atoms each separated by bond length $d=1.4\text{AA}$, and if the ends of the box are a distance $d$ past the end $\mathrm{C}$ atoms, then the length of the box is $\ell =(2N+1)d$. An energy level is occupied by two paired electrons. Suppose the $N$ lowest levels are occupied by electrons, so the wavelength absorption of interest involves the excitation from level $N$ to level $N+1$. Compute the absorption energy $\mathrm{\Delta }\epsilon ={\epsilon }_{N+1}-{\epsilon }_N=hc/\lambda$, where $c$ is the speed of light and $\lambda$ is the wavelength of absorbed radiation, using the particle-in-a-box model.

Heat capacity for ${\mathrm{Cl}}_2.$ What is $C_V$ at $800\mathrm{K}$ for ${\mathrm{Cl}}_2$ treated as an ideal diatomic gas in the high-temperature limit?