Physical Chemistry : Thermodynamics, Statistical Mechanics & Kinetics

Andrew Cooksy

Chapter 9 The Second and Third Laws: Entropy - all with Video Answers

Educators

Chapter Questions

Problem 1

The number of moles in an ideal gas sample is increased adiabatically and at constant volume. During this process, which of the following occurs?
a. $T$ increases and $S$ decreases.
b. $T$ decreases and $S$ increases.
c. $T$ decreases and $S$ remains constant.
d. $T$ and $S$ remain constant.

Alexander Clippinger

Numerade Educator

03:48

Problem 2

Standard entropies for ions in solution are evaluated relative to the entropy of $\mathrm{H}^{+}(\mathrm{aq})$; in other words, $S^{\ominus}\left[\mathrm{H}^{+}(\mathrm{aq})\right]=0$. When $\mathrm{H}_2 \mathrm{~S}$ is added to water, the solution contains $\mathrm{H}_2 \mathrm{~S}(\mathrm{aq}), \mathrm{HS}^{-}(\mathrm{aq})$, and $\mathrm{S}^{2-}(\mathrm{aq})$. The standard entropies (in $\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ) of these three species are written below; write the name of the correct compound next to its entropy.
a. -14.6 ;
b. 62.08 ;
c. 121

Ava Perkins

Numerade Educator

01:51

Problem 3

A rigid container initially at pressure $P_1$ is filled with gas to pressure $P_2$. During this time the temperature is kept constant. Explain whether the entropy of the sample in the container increases or decreases during this process.

Alexander Clippinger

Numerade Educator

04:18

Problem 4

Two containers are at exactly the same temperature, 298.15 K , and hold exactly 0.100 mol of gas, but container A holds helium gas whereas container B holds silane $\left(\mathrm{SiH}_4\right)$ gas. The internal energy of the silane must be higher, because silane has a higher heat capacity than helium. On average, does any energy transfer between the two containers?

Ibrahim Abdullahi

Numerade Educator

01:38

Problem 5

The Ising model lattice is stable with all spins in the same direction for which of the following?
a. high temperature or high external magnetic field
b. low temperature or high external magnetic field
c. high temperature or low external magnetic field
d. low temperature or low external magnetic field

Penny Riley

Numerade Educator

01:33

Problem 6

Calculate the entropy change in $\mathrm{JK}^{-1}$ for heating 1 mole of He gas $\left(C_{\mathrm{Pm}}=20.88 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)$ from 300 K to 400 K at constant pressure.

Sri Datta Vikas Buchemmavari

Numerade Educator

01:16

Problem 7

Calculate the entropy change when a 56 g sample of pure iron is cooled from 373 K to 273 K at a pressure of 1.0 bar. For iron, $C_{P_{\mathrm{m}}}=25.1 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.

Ajay Singhal

Numerade Educator

01:16

Problem 8

Surprisingly, HCCH has a lower standard entropy at 298 K than HCN gas, even though acetylene has more vibrational modes and a lower rotational constant than hydrogen cyanide. For $\mathrm{HCCH}, S_{\mathrm{m}}^{\ominus}=200.9 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, $C_{P \mathrm{~m}}=43.93 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and for $\mathrm{HCN} \mathrm{S} \mathrm{m}_{\mathrm{m}}^{\mathrm{v}}=201.8 \mathrm{~J} \mathrm{~K}^{-1}$, $C_{P \mathrm{~m}}=35.9 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Find the temperature at which HCCH and HCN have the same standard entropy.

Adriano Chikande

Numerade Educator

01:39

Problem 9

Two closed, rigid water containers with initial temperatures $T_{\mathrm{A}}$ and $T_{\mathrm{B}}$ are brought into thermal contact. At equilibrium, the two containers are at the same temperature. Prove that this process has positive $\Delta S$ and find an expression for the final temperature.

Dominador Tan

Numerade Educator

05:24

Problem 10

Find the entropy due to rotation in units of $\mathrm{JK}^{-1}$ of a single molecule in the state $J=5$.

Lewis Rose

Numerade Educator

Problem 11

Engineers sometimes model the propulsion of a bullet or an artillery shell through a gun barrel using the Nobel-Abel equation of state, which is the same as the van der Waals equation but without the attractive $a$ term:

$$
P\left(V_m-b\right)=R T
$$

The justification for this is that the gases involved are very hot (so the attractive term in the intermolecular potential energy is negligible compared to the thermal energy) but extremely compressed (so the excluded volume is a significant fraction of the total volume). Find an expression for the entropy change $\Delta S=S_2-S_1$ in a Nobel-Abel gas as it expands from volume and temperature $V_1, T_1$ to $V_2, T_2$, in terms of the number of moles $n$, the heat capacity $C_V$, and the excluded volume of the gas $b$. Assume that $C_V$ is independent of $T$. THINKING AHEAD $>$ [What parameter can we use to link volume and temperature to $C_V$ ?]

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05:59

Problem 12

A $25.0 \mathrm{~L}, 1.00$ mole sample of air, composed in this case of $75 \% \mathrm{~N}_2$ and $25 \% \mathrm{O}_2$, is separated into its components (by passing it over a liquid nitrogen trap, which condenses the oxygen). The separated $\mathrm{N}_2$ and $\mathrm{O}_2$ are then each allowed to return to the pressure and temperature of the initial air sample. Calculate $\Delta S$.

Shalini Tyagi

Numerade Educator

01:22

Problem 13

Take our expression for the degeneracy of $N$ particles in a box

$$
\Omega=V^N f(E, N)
$$

and find the change in entropy $\Delta S$ if the box expands from volume $V_1$ to $V_2$, keeping the total energy and number of particles constant.

Adriano Chikande

Numerade Educator

01:35

Problem 14

One mole of an ideal monatomic gas is compressed isothermally at temperature $T$ from pressure $P_1$ to $P_2$. What is the change in entropy?

Vishal Gupta

Numerade Educator

00:27

Problem 15

If $\Omega=A V^N$, find $\Delta S=S\left(V_2\right)-S\left(V_1\right)$ in terms of $R$ when $V_2=2 V_1$ and $N=\mathcal{N}_A$.

Erika Bustos

Numerade Educator

02:08

Problem 16

During a reversible adiabatic expansion, one mole of an ideal monatomic gas cools by 100 K , and $\Delta G$ is -1000 J . What is the entropy of the final state in $\mathrm{J} \mathrm{K}^{-1}$ ?

Keshav Singh

Numerade Educator

02:38

Problem 17

In a temperature reservoir B at 298 K , a sample A undergoes an isothermal, isobaric process, for which $\Delta S_{\mathrm{A}}=-327 \mathrm{~J} \mathrm{~K}^{-1}$ and $\Delta H_{\mathrm{A}}=-572 \mathrm{~kJ}$. Calculate the change in entropy of the sample plus its surroundings, $\Delta S_T$ under these conditions, and determine if the process is spontaneous.

Ankur S

Numerade Educator

03:10

Problem 18

When water is frozen at 3000 bar, it can be trapped in any of three crystalline forms, each corresponding to a different state of the water molecule: ice II, ice III, and ice V. Find the apparent molar entropy in $\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ of water at $T=0$ in such a case, assuming that each of these three forms is equally likely.

Banhishikha Sinha

Numerade Educator

08:00

Problem 19

One prediction from the third law of thermodynamics is that the heat capacity $C_p$ near absolute zero must not decrease with increasing temperature. Prove that this is the case by starting from the assumption that over a tiny increment in temperature, we can find some power $x$ such that the heat capacity changes in proportion to $T^x$.

Jack Hou

Numerade Educator

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Problem 20

Calculate the entropy change when we heat calcium chloride from 298 K to 323 K , assuming that the heat capacity remains a constant $72.59 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ over this temperature range.

Ronald Prasad

Numerade Educator

01:59

Problem 21

Find the entropy change $\Delta S$ for a 0.200 mol sample of $\mathrm{N}_2$ heated and compressed from 298 K and 1.00 bar to 373 K and 3.00 bar .

Kratika Bhadauria

Numerade Educator

11:40

Problem 22

We isolate $n$ moles of an ideal gas with heat capacity $C_{P m}$ and then heat the gas and allow it to expand from a state $P_1, V_1, T_1$ to a final state $P_2, V_2, T_2$. Find a general formula for the $\Delta S$ of this process by treating the process as having two steps: an isothermal expansion step and a separate isobaric heating step. Show that the solution does not depend on which step (the expansion or the heating) is carried out first.

Sanjeev Kumar

Numerade Educator

01:18

Problem 23

In adiabatic demagnetization, we use a magnetic field to couple away almost all of the residual energy of a crystal that has been cooled to 2.2 K by liquid helium. (We can get below the normal boiling point of liquid helium by boiling it at lower pressure.) Use Einstein's molar energy of the crystal,

$$
E_{\mathrm{m}}=3 \mathcal{N}_A\langle\varepsilon\rangle_{\mathrm{vib}}=\frac{3 \mathcal{N}_A \omega_E}{e^{\omega / /\left(k_{\mathrm{B}} T\right)}-1}
$$

in the low-temperature limit to estimate the residual molar energy of a crystal with an Einstein frequency of $105 \mathrm{~cm}^{-1}$ at 2.2 K . (The answer is smaller than you may expect.)

Keshav Singh

Numerade Educator

05:42

Problem 24

Figure 8.8 shows the $P$ versus $V$ graphs of three expansions: (a) the reversible isothermal, (b) the irreversible isothermal, and (c) the reversible adiabatic. Plot the corresponding curves for $T$ versus $S$ of these three processes, letting all three processes start from the same initial point ( $T_1, S_1$ ) in the middle of your graph. Be quantitative if possible; otherwise, sketch an approximate curve. Label the curves $\mathrm{a}, \mathrm{b}$, and c . THINKING AHEAD $>[$ How should the areas under these curves compare?]

Khoobchandra Agrawal

Numerade Educator

04:36

Problem 25

Evaluate the molar difference between the free energy and the energy, $G_{\mathrm{m}}-E_{\mathrm{m}}$ for argon at 298 K , assuming it is an ideal gas.

Mukesh Devi

Numerade Educator

02:00

Problem 26

The container shown in the following figure separates an ideal gas into two compartments with a movable, thermally conducting wall between them. The conditions in each compartment are labeled.

Circle each of the following statements that correctly describes how the system will change.
a. the wall will move to the right
b. the wall will move to the left
c. heat will flow to the right
d. heat will flow to the left
e. until the volume on both sides is the same
f. until the temperature on both sides is the same
g. until the pressure on both sides is the same
h. until the number of moles on both sides is the same

Ankur S

Numerade Educator

01:39

Problem 27

Show that the second law of thermodynamics applies to an isolated chamber, with two chambers A and $B$ separated by a rigid, gas-tight piston. Both chambers are filled with equal amounts of the same gas, but gas A is initially at greater pressure than gas B. The piston slides from left to right until the two pressures are the same. Heat can flow freely between chambers A and B, so the process is isothermal. Find the total entropy change $\Delta S_T$ of the system, and show whether the value is always positive or always negative.

Dominador Tan

Numerade Educator

04:05

Problem 28

Use the Sackur-Tetrode equation to predict the standard molar entropy of atomic iodine $\mathrm{I}(\mathrm{g})$ at 573 K and 1.00 bar.

Shubham Kumar

Numerade Educator

01:33

Problem 29

In many systems, the canonical distribution accurately describes populations in each degree of freedom (translations, rotations, and vibrations), but the effective temperature for each degree of freedom is different. For example, plasmas often heat vibrational motions more than they heat rotations or translations. Consider a sample of $\mathrm{I}_2$ gas ( $\omega_e=214.5 \mathrm{~cm}^{-1}, B_e=0.0559 \mathrm{~cm}^{-1}$ ) that initially has a vibrational temperature $T_{\text {vib }}=653 \mathrm{~K}$, a rotational temperature $T_{\text {rot }}=437 \mathrm{~K}$, and a translational temperature $T_{\text {trans }}=298 \mathrm{~K}$. We then isolate the sample and wait for these different degrees of freedom to exchange energy until the temperatures are equal. What is the final temperature (now the same for all motions) of the sample?

Adriano Chikande

Numerade Educator

02:27

Problem 30

Find the entropy of mixing, $\Delta_{\text {mix }} S$, for two substances if they are mixed in the proportions that maximize $\Delta_{\text {mix }} S$.

Chai Santi

Numerade Educator

02:39

Problem 31

Evaluate $\Delta_{\text {mix }} S$ for the mixing of two 1 L containers of water at 300 K . What is the change in $\ln \Omega$ for this process, given that $S_{\mathrm{m}}(298 \mathrm{~K})$ is $69.91 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ?

Ronald Prasad

Numerade Educator

04:09

Problem 32

Calculate $\Delta_{\text {mix }} G$ in $J$ for mixing three moles of helium and two moles of argon gas at 300 K , assuming they are ideal gases.

Mukesh Devi

Numerade Educator

01:40

Problem 33

Calculate $\Delta_{\text {mix }} G$ in J at 298 K for mixing 0.01 mole each of $\mathrm{H}_2, \mathrm{Cl}_2$, and $\mathrm{O}_2$, assuming they are ideal gases.

Kratika Bhadauria

Numerade Educator

03:55

Problem 34

Calculate $\Delta G$ in the preparation of 1000 ml of 2.0 M $\mathrm{HClO}_4$ solution, assuming ideal mixing and that each ion contributes individually to the entropy, at 298 K .

Carina Carlos

Numerade Educator

12:51

Problem 35

The desalinization of water attracts growing interest as potable water sources become more scarce and variable, but the process is not an easy one. Even assuming that the ions did not interact strongly with the solvent, desalinization must fight a substantial uphill battle against entropy. Assuming ideal mixing, calculate the minimum $\Delta S$ in $\mathrm{J} \mathrm{K}^{-1}$ for the obtaining of pure water from 10 L of 1.0 M NaCl solution.
Assume that the NaCl ionizes completely in solution and that each ion contributes independently to the entropy.

Saman Zulfiqar

Numerade Educator

04:09

Problem 36

Calculate the entropy of mixing when we combine 0.100 mol of neon with 0.900 mol of argon at 298 K and 1.00 bar, assuming both are ideal gases.

Mukesh Devi

Numerade Educator

Problem 37

Find an expression for $\Delta_{\text {mix }} F$ for the isothermal and isobaric mixing of two ideal gases.

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01:44

Problem 38

For the isothermal mixing of two ideal substances at constant overall pressure $P$ and volume $V_T=V_{\mathrm{A}}+V_{\mathrm{B}}$, we found

$$
\Delta_{\mathrm{mix}} S=-R\left(n_{\mathrm{A}} \ln X_{\mathrm{A}}+n_{\mathrm{B}} \ln X_{\mathrm{B}}\right) .
$$

If the initial isolated samples A and B are identical ( $n_{\mathrm{A}}=n_{\mathrm{B}}, V_{\mathrm{A}}=V_{\mathrm{B}}$ ), and the total volume $V_T$ is allowed to decrease, what is the final total volume $V_T$ in terms of $V_A$ if the overall process is adiabatic and isothermal?

Anand Jangid

Numerade Educator

01:39

Problem 39

Two ideal gases, $n_A$ moles of A and $n_{\mathrm{B}}$ moles of B , initially occupy distinct volumes $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ at different temperatures, $T_{\mathrm{A}}$ and $T_{\mathrm{B}}$. The molar heat capacities at constant pressure are $C_{\mathrm{A}}$ and $C_{\mathrm{B}} . \mathrm{A}$ and B are combined to make a total volume $V_{\mathrm{A}}+V_{\mathrm{B}}$ at equilibrium. First write an equation for the total $\Delta S$ of this process in terms of the parameters above and the final temperature $T_f$. Then find the equation for $T_f$.

Dominador Tan

Numerade Educator

01:11

Problem 40

Consider the isothermal and isobaric mixing of the contents of two containers A and B holding identical samples of the same non-ideal gas, obeying the virial expansion (Eq. 4.39) to second order: $$P=R T\left[\frac{n}{V}+B_2(T)\left(\frac{n}{V}\right)^2\right]$$
The combined volume of both containers is $V_{\mathrm{A}}+V_{\mathrm{B}}=V$, and the total number of moles of gas is $n_{\mathrm{A}}+n_{\mathrm{B}}=n$, where $V_{\mathrm{A}}=V_{\mathrm{B}}$ and $n_{\mathrm{A}}=n_{\mathrm{B}}$. Find an expression for $\Delta S$ of this process in terms of $n, V$, and $B_2(T)$.

Ajay Singhal

Numerade Educator

05:48

Problem 41

In a molecular dynamics calculation of the gas phase, after each time step the computer calculates new positions and orientations for each molecule. If it carries out one calculation for each translational coordinate and one for each rotational coordinate, how many calculations is this in a simulation of 1000 diatomic molecules over a period of $10^{-8} \mathrm{~s}$ with a resolution (time step-size) of $10^{-14} \mathrm{~s}$ ?

Eduard Sanchez

Numerade Educator

01:40

Problem 42

Consider a molecular dynamics simulation using a square well potential where
$$u_{\mathrm{sq}}(R)=\left\{\begin{array}{cc}\infty & \text { if } R \leq R_1 \\-\varepsilon & \text { if } R_1<R \leq R_2 . \\0 & \text { if } R>R_2\end{array}\right.$$
To correct the motion of the particles, we need to calculate the force of any interaction between two molecules A and B .
a. What is this force when A and B are separated by a distance $2 R_2$ ?
b. What is this force when A and B are separated by a distance $\left(R_1+R_2\right) / 2$ ?
c. Use the change in potential energy to find an approximate expression for this force when the distance between A and B changes from $R_2+(\Delta R / 2)$ to $R_2-(\Delta R / 2)$, where $\Delta R \ll R_2-R_1$.

Carson Merrill

Numerade Educator

08:49

Problem 43

A Monte Carlo simulation is carried out on the rotating dipole moment $\mu$ in an external electric field $\mathcal{E}$, which has potential energy
$$U(\theta)=\mathcal{E} \mu \cos \theta .$$ The following table shows a series of six values of $U / k_{\mathrm{B}}$ obtained during the simulation, whether they are included in the running average or not, and the corresponding values of the random number $y$. The previous energy in the simulation is given as "state $0^{\circ}$ in the table, and the temperature is 300 K . Calculate $\langle U\rangle / k_{\mathrm{B}}$ for this series, using only the correct states from the series.

Amit Srivastava

Numerade Educator

22:43

Problem 44

The following table gives averaged total energies for a series of Monte Carlo Ising model simulations at different

Christina Vaughan

Numerade Educator

16:08

Problem 45

In a Monte Carlo simulation of a pure liquid at temperature $T$, we begin with some distribution of $N$ particles, which interact according to a Lennard-Jones potential $u\left(R_{i j}\right)=4 \varepsilon\left[\left(R_{\mathrm{L} j} / R_{i j}\right)^{12}-\left(R_{\mathrm{L} j} / R_{i j}\right)^6\right]$. We select a molecule $i$ and transfer it to a new location $\vec{r}_i=\left(x_i, y_i, z_i\right)$ in the sample to generate a trial state. A random number $Y$ between 0 and 1 is generated.
a. List the conditions you should test to determine whether the trial state is to be kept in the simulation with another particle.
b. Write an equation for the total force at work on the relocated molecule.

Christopher Dzorkpata

Numerade Educator

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Problem 47

Calculate the total energy in terms of $D$ of the $3 \times 3$ Ising model configuration drawn below, if the energy of interaction is $\pm D$ for any atom and each of its eight nearest neighbors. Include the periodic boundary conditions, which have not been drawn.

Susan Hallstrom

Numerade Educator

02:02

Problem 48

In a $1 \mathrm{~cm}^2$ square layer of metal atoms arranged in a square lattice with lattice constant $2.5 \AA$, what fraction of atoms will be found at the very edge of the lattice?
(This problem justifies the need for elimination of edge effects in the Ising model.)

Keshav Singh

Numerade Educator

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Problem 49

Find the maximum energy in terms of $D$ of the $3 \times 3$ two-dimensional Ising model, and draw one of the configurations with that energy.

Susan Hallstrom

Numerade Educator

Problem 50

In terms of the interaction constant $D$, find the difference in the average energy per atom of a ground state $50 \times 50$ two-dimensional Ising model with and without the use of periodic boundary conditions.

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Problem 51

For the following Ising model lattice, how much energy is necessary to flip the spin of the center atom, if $D=2 \mathrm{eV}$ ?

$$
\begin{aligned}
& -\quad+\quad- \\
& -\quad-\quad- \\
& -\quad+\quad+
\end{aligned}
$$

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08:26

Problem 52

Draw any $4 \times 4$ grid for the Ising model that has zero energy, if periodic boundary conditions are included.

Hafiz Shahzaib

Numerade Educator

Problem 53

Derive a formula for the minimum energy in the two-dimensional Ising model, if the interaction energy between each spin and each of its eight nearest neighbors is $-D$. Assume that the model consists of a rectangular array of $i \times j$ spins.

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Problem 54

A simpler Ising model than the one we have used counts only the interactions with the four nearest neighbor spins (above, below, left, and right) rather than the nearest eight. Apply the periodic boundary conditions and use this simpler system to find the potential energy in eV for the following array of spins, if the interaction constant $D=1.5 \mathrm{eV}$.

$$
\begin{aligned}
& +\quad-\quad- \\
& -\quad+\quad- \\
& +\quad-\quad+
\end{aligned}
$$

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01:40

Problem 55

Shown below is a metastable configuration obtained from the Monte Carlo simulation of an Ising spin model under the condition $D=4 k_{\mathrm{B}} T$. If the simulation proceeds by flipping the circled spin $\oplus$ from + to - , find the probability that that new configuration will be kept.

$$
\begin{aligned}
& +\quad+\quad+\quad-\quad-\quad-\quad- \\
& +\quad+\quad+\quad \oplus \quad-\quad-\quad-\quad- \\
& +\quad+\quad+\quad+\quad-\quad-\quad-\quad-
\end{aligned}
$$

Dominador Tan

Numerade Educator

16:28

Problem 56

If a magnetic spin system has an energy obeying the equation

$$
E=E_0\left\{\left[\left(\frac{T}{T_0}\right)+\sqrt{2}\right]^{-4}-\left[\left(\frac{T}{T_0}\right)+\sqrt{2}\right]^{-2}\right\}
$$

where $E_0$ and $T_0$ are constants, find the temperature (in terms of $T_0$ ) at which the heat capacity $C_V$ reaches its maximum value, given that the maximum is the only critical point.

Dr. Rajveer Singh

Numerade Educator

Problem 446

Add the periodic boundary conditions and calculate the energy in terms of $D$ for the following two-dimensional Ising model:

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