Chapter 10, The Boltzmann Distribution Law Video Solutions, Molecular Driving Forces

Problem 1

Statistical thermodynamics of a cooperative system. Perhaps the simplest statistical mechanical system having 'cooperativity' is the three-level system in Table $10.2$.
Table $10.2$
\begin{tabular}{lccc}
\hline Energies & $2 \varepsilon_{0}$ & $\varepsilon_{0}$ & 0 \\
Degeneracies & $\gamma$ & 1 & 1 \\
\hline
\end{tabular}
(a) Write an expression for the partition function $q$ as a function of energy $\varepsilon$, degeneracy $\gamma$ (see page 178 ), and temperature $T$.
(b) Write an expression for the average energy $\langle\varepsilon\rangle$ versus $T$.
(c) For $\varepsilon_{0} / k T=1$ and $\gamma=1$, compute the equilibrium populations, or probabilities, $p_{1}^{*}, p_{2}^{*}, p_{3}^{*}$ of the three levels of energy $0, \varepsilon_{0}, 2 \varepsilon_{0}$, respectively.
(d) Now if $\varepsilon_{0}=2 \mathrm{kcal} \mathrm{mol}^{-1}$ and $\gamma=1000$, find the temperature $T_{0}$ at which $p_{1}=p_{3}$.
(e) Under condition (d), compute $p_{1}^{*}, p_{2}^{*}$, and $p_{3}^{*}$ at temperature $T_{0}$.

Stanley Enemuo

Numerade Educator

01:29

Problem 2

The speed of sound. The speed of sound in air is approximately the average velocity $\left\langle v_{x}^{2}\right\rangle^{1 / 2}$ of the gas molecules. Compute this speed for $T=0^{\circ} \mathrm{C}$, assuming that air is mostly nitrogen gas.

Adriano Chikande

Numerade Educator

02:58

Problem 3

The properties of a two-state system. Given a twostate system in which the low energy level is $600 \mathrm{cal} \mathrm{mol}^{-1}$, the high energy level is $1800 \mathrm{cal} \mathrm{mol}^{-1}$, and the temperature of the system is $300 \mathrm{~K}$,
(a) What is the partition function $q$ ?
(b) What is the average energy $\langle\varepsilon\rangle$ ?

Suzanne W.

Numerade Educator

11:06

Problem 4

Binding to a surface. Consider a particle that has two states: bonded to a surface, or non-bonded (released). The non-bonded state is higher in energy by an amount $\varepsilon_{0}$.
(a) Explain how the ability of the particle to bond to the surface contributes to the heat capacity, and why the heat capacity depends on temperature.
(b) Compute the heat capacity $C_{V}$ in units of $N k$ if $T=300 \mathrm{~K}$ and $\varepsilon_{0}=1.2 \mathrm{kcal} \mathrm{mol}^{-1}$ (which is about the strength of a weak hydrogen bond in water).

Linda Winkler

Numerade Educator

06:03

Problem 5

Entropy depends on distinguishability. Given a system of molecules at $T=300 \mathrm{~K}, \quad q=1 \times 10^{30}$, and $\Delta U=3740 \mathrm{~J} \mathrm{~mol}^{-1}$,
(a) What is the molar entropy if the molecules are distinguishable?
(b) What is the molar entropy if the molecules are indistinguishable?

Narayan Hari

Numerade Educator

04:01

Problem 6

The Boltzmann distribution of uniformly spaced energy levels. A system has energy levels uniformly spaced at $3.2 \times 10^{-20} \mathrm{~J}$ apart. The populations of the energy levels are given by the Boltzmann distribution.
What fraction of particles is in the ground state at $T=300 \mathrm{~K}$ ?

Nicolas Barroga

Numerade Educator

07:22

Problem 7

The populations of spins in a magnetic field. The
nucleus of a hydrogen atom, a proton, has a magnetic
moment. In a magnetic field, the proton has two states
of different energy: spin up and spin down. This is
the basis of proton NMR. The relative populations can
be assumed to be given by the Boltzmann distribution,
where the difference in energy between the two states is
$\Delta \varepsilon=g \mu B, g=2.79$ for protons, and $\mu=5.05 \times 10^{-24} \mathrm{~J} \mathrm{~T}^{-1}$.
For a $300 \mathrm{MHz} \mathrm{NMR}$ instrument, $B=7 \mathrm{~T}$.
(a) Compute the relative population difference,
|N+ $N_{-} \mid /\left(N_{+}+N_{-}\right)$, at room temperature for a
(b) Describe how the population difference changes
with temperature.
(c) What is the partition function?

Eduard Sanchez

Numerade Educator

01:14

Problem 8

Energy and entropy for indistinguishable particles. Equations (10.36) for $\langle\varepsilon\rangle$ and (10.39) for $S$ apply to distinguishable particles. Compute the corresponding quantities for systems of indistinguishable particles.

Ajay Singhal

Numerade Educator

02:50

Problem 9

Computing the Boltzmann distribution. You have a thermodynamic system with three states. You observe the probabilities $p_{1}=0.9, p_{2}=0.09$, and $p_{3}=0.01$ at $T=300 \mathrm{~K}$. What are the energies $\varepsilon_{2}$ and $\varepsilon_{3}$ of states 2 and 3 relative to the ground state?

Adriano Chikande

Numerade Educator

00:54

Problem 10

The pressure reflects how energy levels change with volume. If energy levels $\varepsilon_{i}(V)$ depend on the volume of a system, show that the pressure is the average
$$
p=-N\left\langle\frac{\partial \varepsilon}{\partial V}\right\rangle .
$$

Dan Ni

Numerade Educator

03:25

Problem 11

The end-to-end distance in polymer collapse. Use the two-dimensional four-bead polymer of Example $10.3$. The distance between the chain ends is 1 lattice unit in the compact conformation, 3 lattice units in the extended conformation, and $\sqrt{5}$ lattice units in each of the other three chain conformations. Plot the average end-to-end distance as a function of temperature if the energy is
(a) $\varepsilon=1 \mathrm{kcal} \mathrm{mol}^{-1}$;
(b) $\varepsilon=3 \mathrm{kcal} \mathrm{mol}^{-1}$.

Ameer Said

Numerade Educator

05:03

Problem 12

The lattice model of dimerization. Use the lattice model for monomers bonding to form dimers, and assume large volumes $V \gg 1$.
(a) Derive the partition function.
(b) Compute $p_{1}(T)$ and $p_{2}(T)$, the probabilities of monomers and dimers as functions of temperature, and sketch the dependence on temperature.
(c) Compute the bond breakage temperature $T_{0}$ at which $p_{1}=p_{2}$.

Khoobchandra Agrawal

Numerade Educator

06:45

Problem 13

Deriving the Boltzmann law two different ways. Use Equation (10.6) to show that the distribution of probabilities $p_{j}^{*}$ that minimizes the free energy $F$ at constant $T$ is the same one you get if instead you maximize the entropy $S$ at constant $U=\langle E\rangle$.

Chris Trentman

Numerade Educator

02:21

Problem 14

Protein conformations. Assume a protein has six different discrete conformations, with energies given in Figure 10.13.
A. $-5 \mathrm{kcal} \mathrm{mol}^{-1}$
B. $-3 \mathrm{kcal} \mathrm{mol}^{-1}$
C. $-3 \mathrm{kcal} \mathrm{mol}^{-1}$
D. $1 \mathrm{kcal} \mathrm{mol}^{-1}$
E. $1 \mathrm{kcal} \mathrm{mol}^{-1}$
F. $2 \mathrm{kcal} \mathrm{mol}^{-1}$
Figure $10.13$
(a) Write an expression for the probability $p(i)$ of finding the protein in conformation $i$.
(b) Write an expression for the probability $p(E)$ of finding the protein having energy $E$.
(c) Use the expressions you wrote in (a) and (b) to calculate the following probabilities:
(i) $p$ (State B).
(ii) $p$ (State A).
(iii) $p$ (State D).
(iv) $p\left(1 \mathrm{kcal} \mathrm{mol}^{-1}\right)$.
(v) $p\left(-5 \mathrm{kcal} \mathrm{mol}^{-1}\right)$.
(d) What is the average energy of the ensemble of conformations?

Rabeya Zahid

Numerade Educator

00:31

Problem 15

Modeling ligand binding. A ligand is bound to a protein with a spring-like square-law energy $\varepsilon(x)$, where $x$ is the distance between the ligand and protein as shown in Figure 10.14.
$$
\varepsilon(x)=\frac{1}{2} c x^{2}
$$
Protein
Figure $10.14$
(a) For constant $(T, V, N)$, write an expression for the probability distribution $p(x)$ of the ligand separation from the protein.
(b) Sketch a plot of $p(x)$ versus $x$.
(c) Write an expression for the average location of the ligand, $\langle x\rangle$.
(d) Write an expression for the second moment of the location of the ligand, $\left\langle x^{2}\right\rangle$.
(e) Calculate the average energy $\langle\varepsilon\rangle$ of the system.

Shazia Naz

Numerade Educator

01:18

Problem 16

Distribution of torsional angles. In a computer simulation that samples a molecular torsion angle $\theta$, you observe a Gaussian distribution $p(\theta)$, shown in Figure 10.15:
$$
p(\theta)=p_{0} e^{-k_{s}\left(\theta-\theta_{0}\right)^{2}} .
$$
Figure $10.15$
What is the underlying energy function $E(\theta)$ that gives rise to it?

Averell Hause

Carnegie Mellon University

01:40

Problem 17

Three-bead polymer chain model. Consider a three-bead polymer that can undergo conformational change from a nonlinear to a linear form, as shown in Figure $10.16$. Both conformations have the same energy. Now suppose the $X$ and $Y$ atoms of the polymer can bind a ligand $L$ (Figure 10.17). Breaking one bond increases the energy by $\varepsilon$ and breaking two bonds increases the energy by $2 \varepsilon$. Assume that the ligand-bound conformation has the lowest energy.
Figure $10.16$
(a) Draw a picture showing the possible binding states.
(b) Calculate the equilibrium probabilities of these conformations.
Figure $10.17$
(c) Plot the population distribution of the conformations at temperatures $k T=0$ and $k T=1.4 \varepsilon$. What is the difference in entropy between those two temperatures?
(d) Plot the distribution at high temperatures $(T \rightarrow \infty)$ and explain its shape.
(e) What is the average energy of the system at each temperature in part (c)?

Carson Merrill

Numerade Educator

03:26

Problem 18

Protein hinge motions. A protein has two domains, connected by a flexible hinge as shown in Figure $10.18$.
Figure $10.18$
The hinge fluctuates around an angle $\theta_{0}$. The distribution of the angle $\theta$ is Gaussian, around $\theta_{0}$, as shown in Figure 10.19:
$$
p(\theta)=\left(\frac{1}{\sigma \sqrt{2 \pi}}\right) \exp \left[-\frac{\left(\theta-\theta_{0}\right)^{2}}{2 \sigma^{2}}\right],
$$
where $\sigma$ is the standard deviation.
(a) Assume a Boltzmann distribution of probabilities: $p(\theta)=b \exp \left[-c\left(\theta-\theta_{0}\right)^{2} / k T\right]$, with energy $\varepsilon=c\left(\theta-\theta_{0}\right)^{2}$. Derive the value of $b$.
(b) If you measure a standard deviation $\sigma=30^{\circ}$, what is the value of $c$ ? Assume $T=300 \mathrm{~K}$.
Figure 10.19
(c) Derive an expression relating $\left\langle\theta^{2}\right\rangle$ to $c$.
(d) Derive an expression for the entropy in terms of $k T$ and $c: S / k=-\int_{-\infty}^{\infty} p(\theta) \ln [p(\theta)] d \theta$.
(e) If a mutant of the protein has fluctuations that are smaller than the wild type, $\left\langle\theta^{2}\right\rangle_{M}=(1 / 2)\left\langle\theta^{2}\right\rangle_{W T}$, what is the change in entropy, $\Delta S=S_{M}-S_{W T}$, where $S_{M}$ is the entropy of the mutant and $S_{W T}$ is the entropy of the original wild-type protein?

Crystal Wang

Numerade Educator