Molecular Driving Forces

K.Dill and S.Bromberg

Chapter 11 The Statistical Mechanics of Simple Gases & Solids - all with Video Answers

Educators

Chapter Questions

02:07

Problem 1

The heat capacity of an ideal gas. What is the heat capacity $C_{V}$ of an ideal gas of argon atoms?

Vivek Singh

Numerade Educator

04:12

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.

Sachin Rao

Numerade Educator

04:50

Problem 3

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 $10 .$ )

Robert Zaballa

Numerade Educator

02:09

Problem 4

The statistical mechanics of an electron. Calculate the two lowest energy levels for an electron in a box of volume $V=1 A^{3}$ (this is an approximate model for the hydrogen atom). Calculate the partition function at $T=300 \mathrm{~K}$. Are quantum effects important?

Crystal Wang

Numerade Educator

03:37

Problem 5

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.

Adriano Chikande

Numerade Educator

01:26

Problem 6

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?

Penny Riley

Numerade Educator

09:44

Problem 7

The statistical thermodynamics of harmonic oscillations. Write the internal energy, entropy, enthalpy, free energy, and pressure for a system of $N$ independent distinguishable harmonic oscillators.

Tianyu Li

Numerade Educator

19:29

Problem 8

Orbital steering in proteins. To prove that proteins do not require 'orbital steering'' a process once proposed to orient a substrate with high precision before binding. $T$ Bruice has calculated the dependence of the total energy on the rotational conformation of the hydroxymethylene group of 4 -hydroxybutyric acid at $T=300 \mathrm{~K}$. Assume that the curve in Figure $11.17$ is approximately parabolic, $\varepsilon=$ $(1 / 2) k_{s}\left(\alpha-\alpha_{0}\right)^{2}$, where $\alpha$ is the dihedral angle of rotation. Use the equipartition theorem.
Figure $11.17$ Source: TC Bruice, Cold Spring Harbor Symposia on Quantitative Biology 36, 21-27 (1972).
(a) Determine the spring constant $k_{s-}$
(b) What is the average energy $\{\varepsilon\rangle ?$
(c) What is the rms dihedral angle $\left\langle\alpha^{2}\right\rangle^{1 / 2}$ ?

Hafiz Shahzaib

Numerade Educator

02:11

Problem 9

The entropy of crystalline carbon monoxide at $T=0 \mathrm{~K} . \quad$ 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 $T=0 \mathrm{~K}$.

Jennifer Stoner

Numerade Educator

01:23

Problem 10

Temperature-dependent quantities in statistical thermodynamics. Which quantities depend on temperature?
(a) Planck's constant $h$.
(b) Partition function $q$.
(c) Energy levels $\varepsilon_{j}$.
(d) Average energy $\langle\varepsilon\rangle$.
(e) Heat capacity $C_{V}$ for an ideal gas.

Ajay Singhal

Numerade Educator

02:05

Problem 11

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?

Suzanne W.

Numerade Educator

03:56

Problem 12

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 $R=1.128 \times 10^{-10} \mathrm{~m}$.

Susan Hallstrom

Numerade Educator

04:15

Problem 13

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 $2 N$ carbon atoms each separated by bond length $d=1.4 \AA$, and if the ends of the box are a distance $d$ past the end $\mathrm{C}$ atoms, then the length of the box is $\ell=(2 N+1) d$. 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 $\Delta \varepsilon=\varepsilon_{N+1}-\varepsilon_{N}=h c / \lambda$, where $c$ is the speed of light and $\lambda$ is the wavelength of absorbed radiation, using the particle-in-a-box model.

Penny Riley

Numerade Educator

04:15

Problem 14

Why are conjugated bonds so stiff? As in problem 13, model polyene chain boxes of length $\ell \approx 2 N d$, where $d$ is the average length of each carbon-carbon separation, and $2 N$ is the number of carbons. There are $2 N$ electrons in $N$ energy levels, particles distributed throughout 'boxes,' according to the Pauli principle, with at most two electrons per level.
(a) Compute the total energy.
(b) Compute the total energy if the chain is 'bent,' that is, if there are two boxes, each of length $\ell / 2$ containing $N$ electrons each.

Penny Riley

Numerade Educator

01:59

Problem 15

Electrons flowing in wires carry electrical current. Consider a wire $1 \mathrm{~m}$ long and $10^{-4} \mathrm{~m}^{2}$ in cross-sectional area. Consider the wire to be a box, and use the particle-ina-box model to compute the translational partition function of an electron at $T=300 \mathrm{~K}$.

Chai Santi

Numerade Educator

09:29

Problem 16

Fluctuations. A stable state of a thermodynamic system can be described by the free energy $G(x)$ as a function of the degree of freedom $x$. Suppose $G$ obeys a square law, with spring constant $k_{s}, G(x) / k T=k_{s} x^{2}$, as shown in Figure $11.18$.
Figure $11.18$
(a) Compute the mean-square thermal fluctuations $\left\langle x^{2}\right\rangle$ in terms of $k_{s}$.
(b) Some systems have two single minima, with large spring constants $k_{1}$, and others have a single broad minimum with small spring constant $k_{2}$, as shown in Figures 11.19(a) and (b). For example, two-state equilibria may have two single minima, and the free energies near critical points have a single broad minimum. If $k_{2}=(1 / 4) k_{1}$, what is the ratio of fluctuations $\left\langle x_{2}^{2}\right\rangle /\left\langle x_{1}^{2}\right\rangle$ for individual energy wells?
(a) Two Single Minima
(b) One Broad Minimum
$G(x)$
Figure $11.19$

Averell Hause

Carnegie Mellon University

03:46

Problem 17

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?

Keshav Singh

Numerade Educator

03:07

Problem 18

Protein in a box. Consider a protein of diameter $40 \AA$ trapped in the pore of a chromatography column. The pore is a cubic box, $100 \AA$ on a side. The protein mass is $10^{4} \mathrm{~g} \mathrm{~mol}^{-1}$. Assume the box is otherwise empty and $T=300 \mathrm{~K}$.
(a) Compute the translational partition function. Are quantum effects important?
(b) If you deuterate all the hydrogens in the protein and increase the protein mass by $10 \%$, does the free energy increase or decrease?
(c) By how much?

Ahmed Ali

Numerade Educator

03:58

Problem 19

Vibrational partition function of iodine. Compute the value of the vibrational partition function for iodine, $\mathrm{I}_{2}$, at $T=308 \mathrm{~K}$. (Hint: see Table 11.2.)

Shubham Kumar

Numerade Educator

03:27

Problem 20

Electron in a quantum-dot box. An electron moving through the lattice of a semiconductor has less inertia than when it is in a gas. Assume that the effective mass of the electron is only $10 \%$ of its actual mass at rest. Calculate the translational partition function of the electron at room temperature $(273 \mathrm{~K})$ in a small semiconductor particle of a cubic shape with a side
(a) $1 \mathrm{~mm}\left(10^{-3} \mathrm{~m}\right)$,
(b) $100 \AA\left(100 \cdot 10^{-10} \mathrm{~m}\right)$;
(c) To which particle would the term 'quantum dot,' i.e., a system with quantum mechanical behavior, be applied, and why?

Robert Zaballa

Numerade Educator

01:49

Problem 21

A protein, quantum mechanics, and the cell. Assume that a protein of mass $50,000 \mathrm{~g} \mathrm{~mol}^{-1}$ can freely move in the cell. Approximate the cell as a cubic box $10 \mu \mathrm{m}$ on a side.
(a) Compute the translational partition function for the protein in the whole cell. Are quantum effects important?
(b) The living cell, however, is very crowded with other molecules. Now assume that the protein can freely move only $5 \AA$ along each $x, y$, and $z$ direction before it bumps into some other molecule. Compute the translation partition function and conclude whether quantum mechanical effects are important in this case.
(c) Now assume that we deuterate all the hydrogens in the protein (replace hydrogens with deuterium atoms). If the protein mass is increased by $10 \%$, what happens to the free energy of the modified protein? By how much does it change?

M S

Numerade Educator

07:23

Problem 22

Electron in benzene. Consider an electron that can move freely throughout the aromatic orbitals of benzene. Model the electron as a particle in a two-dimensional box $4 \AA \times 4 \AA$.
(a) Compute $\Delta \varepsilon$, the energy change from the ground state to the excited state, $n_{x}=n_{y}=2$.
(b) Compute the wavelength $\lambda$ of light that would be absorbed in this transition, if $\Delta \varepsilon=h c / \lambda$, where $h$ is Planck's constant and $c$ is the speed of light.
(c) Will this transition be in the visible part of the electromagnetic spectrum (i.e., is liquid benzene colored or transparent), according to this simple model?

Edward Zhang

Numerade Educator

02:18

Problem 23

Vibrations in insulin. What is the average energy stored in the vibrational degrees of freedom of one molecule of insulin, a protein with $\mathrm{C}_{256} \mathrm{H}_{381} \mathrm{~N}_{65} \mathrm{O}_{76} \mathrm{~S}_{6}$, at room temperature?

Madi Sousa

Numerade Educator

01:53

Problem 24

Escape velocity of gases from the Moon. The escape velocity of an object to leave the Moon is $3.4 \mathrm{~km} \mathrm{~s}^{-1}$. The temperature on the sunny surface of the Moon is $400 \mathrm{~K}$. What is the weight of the gas that will escape the Moon with an average velocity $\left\langle V_{x}^{2}\right\rangle^{1 / 2}$. What do you conclude about the Moon's atmosphere?

Ronald Prasad

Numerade Educator