An Introduction to Thermodynamics and Statistical Mechanics

Keith Stowe

Chapter 8 Entropy and thermal interactions - all with Video Answers

Educators

Chapter Questions

Problem 1

(a) For a monatomic ideal gas of $N$ molecules, $\Omega(E)$ is proportional to $E$ raised to what power?
(b) Using equation 8.1 show that, for a monatomic ideal gas of $N$ molecules, $E=(3 / 2) N k T$.
(c) Starting with $\Omega(E)=$ constant $\times\left(E_{\text {therm }}\right)^{\alpha N v}$ (equations 6.7 and 6.10 ), where $\alpha$ is any number, use the definitions of entropy and temperature to derive the relationship between $E_{\text {them }}$ and $T$. (Assume that $E=E_{\text {therm }}$.)

Chai Santi

Numerade Educator

06:30

Problem 2

Suppose that there are $10^{28}$ diatomic air molecules in a room.
(a) How many degrees of freedom does this system have?
(b) What is the internal energy of this system at room temperature $(295 \mathrm{~K})$ ?

Sheh Lit Chang

University of Washington

04:34

Problem 3

Two small systems, $A_1$ and $A_2$, are in thermal equilibrium. The number of states accessible to each increases with its energy according to $\Omega_1= \left(E_1 / C\right)^{10}$ and $\Omega_2=\left(E_2 / C\right)^8$, where $C=10^{-23} \mathrm{~J}$. The total energy of the combined system is fixed at $E_0=E_1+E_2=10^{-18} \mathrm{~J}$.
(a) How many degrees of freedom have systems $A_1$ and $A_2$, respectively?
(b) Use the fact that $\partial \Omega_0 / \partial E_1=0$ when $\Omega_0$ is a maximum to find $E_1$ and $E_2$ when the combined system is in equilibrium.
(c) What is the entropy of the combined system in equilibrium?
(d) Using the definition of temperature, and the fact that in equilibrium the temperature of either system is the same, find the temperature of the system.

Jerrah Biggerstaff

Numerade Educator

08:55

Problem 4

Suppose that you don't like the way in which the temperature scale is defined, and you wish to define a scale that gives a nice round number for the value of Boltzmann's constant. You measure your temperatures on this scale in ${ }^{\circ} \mathrm{R}$ (for "degrees round"). In units of ${ }^{\circ} \mathrm{R}$, what would be the boiling point of water if:
(a) $k=1.0 \times 10^{-16} \mathrm{erg} /{ }^{\circ} \mathrm{R}$ ?
(b) $k=10^{-4} \mathrm{eV} /{ }^{\circ} \mathrm{R}$ ?
(c) $k=1.0 \mathrm{~J} /{ }^{\circ} \mathrm{R}$ ?
(d) $k=1.0 \mathrm{eV} /{ }^{\circ} \mathrm{R}$ ?

Deborah Israel

Numerade Educator

01:18

Problem 5

What would be the value of Boltzmann's constant if the temperature of the triple point ( 273.16 K ) were defined as 100 K ? What would be the boiling point of water on this scale?

Prachita Kush

Numerade Educator

05:14

Problem 6

For a monatomic ideal gas, each molecule has three translational degrees of freedom only. Suppose you calibrate your temperature scale by saying that
300 K is defined to be the temperature of a mole of this ideal gas when it has a thermal energy of 3740 J . What would be the value of Boltzmann's constant, $k$ ?

Kyle Godbey

Numerade Educator

03:38

Problem 7

(a) If you add 20 J of heat to a system at $-20^{\circ} \mathrm{C}$, what is the change in its entropy?
(b) By what factor does the number of states accessible to the system increase?

Keshav Singh

Numerade Educator

03:11

Problem 8

(a) How many joules of heat energy would you have to add to the Pacific Ocean (average temperature $T=4^{\circ} \mathrm{C}$, volume $V=0.70 \times 10^9 \mathrm{~km}^3$ ) to double the number of states accessible to it?
(b) Would your answer be the same if you were dealing with a cup of water at $4^{\circ} \mathrm{C}$ instead?

Shahab Ullah

Numerade Educator

09:01

Problem 9

Consider some ice at $-1{ }^{\circ} \mathrm{C}$ in a glass of water at $+10^{\circ} \mathrm{C}$. For each joule of heat energy that flows from the water to the ice, find the change in entropy of (a) the ice, (b) the water, (c) the total system.

Linda Winkler

Numerade Educator

09:01

Problem 10

A $10^{-2} \mathrm{~kg}$ ice cube, initially at $0^{\circ} \mathrm{C}$, melts in the Atlantic Ocean, where the water temperature is $10^{\circ} \mathrm{C}$. After melting, the ice melt heats up to match its $10^{\circ} \mathrm{C}$ environment. (The latent heat of fusion $=333 \mathrm{~J} / \mathrm{g}$ and the specific heat $=4.18 \mathrm{~J} /(\mathrm{g} \mathrm{K})$. Find the change in entropy of
(a) the water that was originally in the ice cube,
(b) the Atlantic Ocean.
(c) By what factor does the number of states available to the combined system change?

Linda Winkler

Numerade Educator

03:50

Problem 11

The number of states accessible to an ideal gas having energy in the range between $E$ and $E+\delta E$ is given by $\Omega_0=$ constant $\times V^N E^{3 N / 2} \delta E$, where $V$ is the volume of the gas and $N$ the number of molecules.
(a) Using equation 8.7 , show that $p V=N k T$.
(b) This is sometimes written as $p V=n R T$, where $n$ is the number of moles of the gas and $R$ is called the "gas constant." What is $R$ in terms of Boltzmann's constant and Avogadro's number?

Anand Jangid

Numerade Educator

Problem 12

The number of states is expressed as a function of various parameters for three systems below. For each, find an "equation of state," which gives the relationship between $p, V, N$, and $T$. ( $C$ and $b$ are constants.)
(a) $\Omega=C \mathrm{c}^{b N V^2}(E V)^N$,
(b) $\Omega=\frac{\pi}{2}\left(\frac{2}{h}\right)^{3 N} V^{N / 2} \mathrm{e}^{b N V} E^{2 N}$,
(c) $\Omega=C \mathrm{e}^{-b / V^{10}} E^{3 N}$.

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07:54

Problem 13

For each system in problem 12 , find the dependence of the internal energy $E$ on $V, N$, and $T$.

Khoobchandra Agrawal

Numerade Educator

Problem 14

For each system in problem 12 , find the dependence of the chemical potential $\mu$ on $E, V, N$, and $T$.

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03:10

Problem 15

Consider two gaseous systems interacting mechanically and thermally but not diffusively. They are isolated from the rest of the Universe, and their total volume is fixed at $V_0$.
(a) Show that the total number of accessible states, $\Omega_0$, is a very sensitive function of the distribution of volume between them. (See equation 6.11 and the argument preceding it for the dependence of $\Omega$ on $V$.)
(b) Since $d V_1=-d V_2$, show that $\partial S_1 / \partial V_1=\partial S_2 / \partial V_2$ when the systems are in equilibrium.
(c) From this, what can you conclude about how $p_1, T_1, p_2$, and $T_2$ are related in equilibrium?

Manish Jain

Numerade Educator

02:33

Problem 16

A hundred grams of water are heated from $10^{\circ} \mathrm{C}$ to $95^{\circ} \mathrm{C}$. If the specific heat of water is $4.19 \mathrm{~J} /\left(\mathrm{g}^{\circ} \mathrm{C}\right)$, what is the increase in entropy of the water during this process?

Chitra Gondi

Numerade Educator

01:10

Problem 17

Three hundred grams of aluminum are heated from $-50^{\circ} \mathrm{C}$ to $300^{\circ} \mathrm{C}$. If the specific heat of aluminum is $0.88 \mathrm{~J} /\left(\mathrm{g}^{\circ} \mathrm{C}\right)$, what is the change in entropy of the aluminum?

Averell Hause

Carnegie Mellon University

Problem 18

Consider the differentials listed below. For each, state whether it is exact. For those that are not exact, find a multiplicative factor $f(x, y)$ such that $f(x, y) \mathrm{d} F$ is exact:
(a) $\mathrm{d} F=2 x \mathrm{~d} x+\left(x^2 / y\right) \mathrm{d} y$,
(b) $\mathrm{d} F=2 x y \mathrm{~d} x+x^2 \mathrm{~d} y$,
(c) $\mathrm{d} F=2 x y^2 \mathrm{~d} x+x^2 y \mathrm{~d} y$,
(d) $\mathrm{d} F=3 x^3 y^2 \mathrm{~d} x+x^4 y \mathrm{~d} y$,
(e) $\mathrm{d} F=\mathrm{d} x / x+\mathrm{d} y / y$,
(f) $\mathrm{d} F=p \mathrm{~d} V+V \mathrm{~d} p$,
(g) $\mathrm{d} F=p^2 \mathrm{~d} V+p V \mathrm{~d} p$.

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06:30

Problem 19

Estimate the total thermal energy of the following systems at 290 K :
(a) the air in your bedroom (one mole occupies 22.4 liters at $0{ }^{\circ} \mathrm{C}$ );
(b) the iron atoms in a 5 gram nail (the atomic mass number for iron is 56);
(c) A diamond of mass 0.1 gram (the atomic mass number of carbon is 12 );
(d) the Pacific Ocean $\left(0.7 \times 10^9 \mathrm{~km}^3\right.$ of water, each molecule having six degrees of freedom).

Sheh Lit Chang

University of Washington

Problem 20

A system has $10^{25}$ degrees of freedom and initial volume $1 \mathrm{~m}^3$ and is under a pressure of $10^5 \mathrm{~N} / \mathrm{m}^2$. While held at a constant temperature (assume constant internal energy) of $17^{\circ} \mathrm{C}$, it expands by $1 \mathrm{~mm}^3$.
(a) Does the number of accessible states increase or decrease?
(b) By how many times?

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02:13

Problem 21

In a certain system, the number of accessible states increases by a factor $10^{10^{20}}$ when 1 joule of energy is added at constant $V$ and $N$.
(a) What is the increase in entropy, $\Delta S$ ?
(b) What is the temperature of the system?

Aparna Shakti

Numerade Educator

10:44

Problem 22

Which of the following differentials, $\mathrm{d} E, \mathrm{~d} V, \mathrm{~d} Q, \mathrm{~d} p, \mathrm{~d} T, \mathrm{~d} \mu, \mathrm{~d} N, \mathrm{~d} W, \mathrm{~d} S$, are exact? For each that is inexact, find a multiplicative factor involving $E, V, T, p, N$, etc. that would make it exact.

Eduard Sanchez

Numerade Educator

03:04

Problem 23

Suppose that the heat entering a system can be expressed in terms of the temperature and volume as $d Q=b\left(T^2 d V+T V \mathrm{~d} T\right)$ and that the work done by the system can be expressed in terms of its pressure and volume as $\mathrm{d} W= c\left(4 p^4 V^2 \mathrm{~d} p+2 p^5 V \mathrm{~d} V\right)$, where $b$ and $c$ are constants. Find the multiplicative factors that turn these inexact differentials into exact ones.

Lottie Adams

Numerade Educator

04:39

Problem 24

Consider a system of $10^{25}$ particles at temperature 295 K , pressure $10^5 \mathrm{~Pa}$, and chemical potential -0.3 eV . It experiences the following very small changes: $2.2 \times 10^{-2} \mathrm{~J}$ of heat are added, it expands by $10^{-7} \mathrm{~m}^3$ and it gains $10^{17}$ particles. What is the change in its (a) internal energy, (b) entropy?

Surendra Kumar

Numerade Educator

02:13

Problem 25

Consider a system at 300 K to which 1 joule of heat is added.
(a) What is its change in entropy?
(b) By what factor does the number of accessible states increase?

Aparna Shakti

Numerade Educator

Problem 26

For any system, the following quantities are all interrelated: the number of degrees of freedom, $N v$, the change in entropy, $\Delta S$, the ratio of final to initial thermal energies, $E_{\mathrm{f}} / E_{\mathrm{i}}$, and the ratio of final to initial states, $\Omega_{\mathrm{f}} / \Omega_{\mathrm{i}}$. Below is a table, each horizontal row representing some process for some system. Can you fill in the missing numbers?
( table can't copy )

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02:24

Problem 27

(a) By how much does the entropy of the Atlantic Ocean ( $T=280 \mathrm{~K}, V= 0.36 \times 10^9 \mathrm{~km}^3$ ) change when 0.1 joule of heat is added?
(b) What about a cup of water at 280 K ?
(c) By what factor does the number of states available to each system increase?

Hubert Agamasu

Numerade Educator

05:07

Problem 28

A system of $10^{24}$ particles with $\mu=-0.2 \mathrm{eV}$ is at room temperature (295 $\mathrm{K})$. Find the factor by which the number of accessible states increases in the following cases.
(a) The number of particles is increased by $0.01 \%$, without adding energy to, or doing work on, the system. (That is, the incoming particles have zero total energy $u_0+\varepsilon_{\text {therm }}=0$, and the system's volume is unchanged.)
(b) A single energyless particle is added.

Eduard Sanchez

Numerade Educator

14:32

Problem 29

A rubber ball is in contact with a heat reservoir that keeps its temperature constant at 300 K . (So assume that the internal energy is constant.) At a pressure of 1.001 atm its volume decreases by $10^{-3} \mathrm{~cm}^3$.
(a) What is the change in entropy?
(b) By what factor does the number of accessible states change?
(c) Repeat for a temperature of $20^{\circ} \mathrm{C}$, a pressure of $1.02 \times 10^5 \mathrm{~Pa}$, and a volume reduction of $10^{-10} \mathrm{~m}^3$.

David Morabito

Numerade Educator

Problem 30

A magnet is in contact with a reservoir that keeps it at 300 K . The magnet has a magnetic moment $\mu_z=10^{-3} \mathrm{~J} / \mathrm{T}$ and is sitting in an external field oriented along the $z$-axis of strength $B_z=0.1 \mathrm{~T}$. The external field is increased by $1 \%$, and the induced magnetic moment also increases by $1 \%$.
(a) What is the change in entropy?
(b) By what factor does the number of accessible states change?

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02:13

Problem 31

By how much does the entropy of a system increase if the number of accessible states doubles?

Aparna Shakti

Numerade Educator

03:23

Problem 32

Consider $1 \mathrm{~m}^3$ of steam held at a temperature of $600{ }^{\circ} \mathrm{C}$ and a pressure of 50 atm . By how much must its volume expand if the number of accessible states increases a billionfold?

Michael Mackenzie

Numerade Educator

02:20

Problem 33

Consider an insulated ideal gas, such that no heat energy can enter or leave. It is slowly compressed to $96 \%$ of its original volume, and its temperature rises correspondingly.
(a) What is the change in its entropy?
(b) An increase in temperature indicates more thermal motion and larger momenta. That would imply that the particles have more accessible room in momentum space with a corresponding increase in number of accessible quantum states. How, then, can you justify your answer to part (a)?

Satpal Satpal

Numerade Educator

02:13

Problem 34

When one joule of heat energy is added to a system, the number of accessible states increases by a factor of $10^{10^{10}}$. What is the temperature of the system?

Aparna Shakti

Numerade Educator

02:28

Problem 35

You have a system of $10^{24}$ particles at 300 K . If you hold the volume constant and add $10^{19}$ more perfectly energyless particles, you notice the temperature
of the system rises slightly, as if you had added 0.4 J of heat. Estimate the chemical potential for the particles of this system (in eV ).

Khoobchandra Agrawal

Numerade Educator

00:53

Problem 36

Why can't heat capacities be defined at ordinary phase transitions?

Rashmi Gondi

Numerade Educator

04:13

Problem 37

Calculate the change in entropy for 0.24 kg of each of the following as it is heated from 295 K to 296 K (use Table 8.1): (a) Water, (b) gold, (c) marble, (d) wood.

Salamat Ali

Numerade Educator

01:17

Problem 38

Which of the above materials would make the most efficient reservoir for the storage of solar heat? Why do you suppose daily and seasonal temperature changes are greater inland than on the coast?

Evey Z

Numerade Educator

00:52

Problem 39

For a material that expands as its temperature rises, do you think $C_p$ or $C_V$ would be larger? Why?

Lottie Adams

Numerade Educator

03:10

Problem 40

(a) One helium atom has mass $6.7 \times 10^{-27} \mathrm{~kg}$. Estimate the specific heat at constant volume $c_V$ for helium in $\mathrm{kJ} /(\mathrm{kg} \mathrm{K})$. (Note that if the volume is constant, no work is done as the heat is added.)
(b) Why does your answer differ from that in Table 8.1?

Eduard Sanchez

Numerade Educator

Problem 41

For an ideal gas having a fixed number $N$ of particles, the internal energy depends only on the temperature and not at all on the volume: $E=(v / 2) N k T$. Furthermore, the pressure, volume, and temperature are related by the ideal gas law, $p V=N k T$. The first law for a system of a fixed number of particles reads $\mathrm{d} E=\mathrm{d} Q-p \mathrm{~d} V$. Using this information and the definition of molar heat capacities, show that (a) $C_V=(v / 2) R$, (b) $C_p=C_V+R$, where $R= N_A k$ is the molar gas constant.

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00:31

Problem 42

(a) If 2000 J of heat are added to 100 g of gold, initially at $0^{\circ} \mathrm{C}$, what is the change in entropy?
(b) By what factor does the number of accessible states increase?

David Collins

Numerade Educator

01:20

Problem 43

Repeat the above problem for 500 g of aluminum.

Lottie Adams

Numerade Educator

02:14

Problem 44

Compute the change in entropy of 1 gram of water as it goes from solid ice at $0^{\circ} \mathrm{C}$ to water vapor at $100^{\circ} \mathrm{C}$. The latent heat of fusion is $330 \mathrm{~J} / \mathrm{g}$ and of vaporization is $2260 \mathrm{~J} / \mathrm{g}$.

Ajay Singhal

Numerade Educator

02:58

Problem 45

Consider the chemical reaction $2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{C}$. You produce 1 mole of C at 500 K and atmospheric pressure. The molar heat capacities of these substances at atmospheric pressure are all zero between 0 K and 10 K and constant above 10 K , being given for $T>10 \mathrm{~K}$ by

$$
\begin{gathered}
C_{\mathrm{A}}=19.4 \mathrm{~J} /(\text { mole K }), \quad C_{\mathrm{B}}=35.9 \mathrm{~J} /(\text { mole K }), \\
C_{\mathrm{C}}=67.7 \mathrm{~J} /(\text { mole K }) .
\end{gathered}
$$
(a) How much heat will be released in this reaction?
(b) How can the system lose entropy and not violate the second law?

Zubair Abdulla

Numerade Educator

02:58

Problem 46

Consider the chemical reaction $2 \mathrm{~A}+2 \mathrm{~B} \rightarrow \mathrm{C}+\mathrm{D}$. You produce 1 mole of C at 300 K and atmospheric pressure. How much heat will be released in this reaction if the molar heat capacities of these substances, in units of $\mathrm{J} /$ (mole K ), at atmospheric pressure are as follows?:

$$
\begin{array}{ll}
C_{\mathrm{A}}=18.6(T / 200 \mathrm{~K})^{1 / 3}, & C_{\mathrm{B}}=15.7(T / 200 \mathrm{~K})^{1 / 3}, \\
C_{\mathrm{C}}=23.4(T / 200 \mathrm{~K})^{1 / 3}, & C_{\mathrm{D}}=39.7(T / 200 \mathrm{~K})^{1 / 3} .
\end{array}
$$

Zubair Abdulla

Numerade Educator

01:36

Problem 47

The latent heat of fusion for iron at 1809 K is $246 \mathrm{~J} / \mathrm{g}$, or $13790 \mathrm{~J} / \mathrm{mole}$. Its atomic mass number is 56 . In the solid state, each iron atom has six degrees of freedom, and in the liquid state each has only three.
(a) What is the change in the potential energy reference level, $u_0$, when iron goes from solid to liquid?
(b) What is the change in entropy per gram?

David Collins

Numerade Educator

01:10

Problem 48

Repeat the above problem, but for the sublimation of dry ice. $\mathrm{CO}_2$ has a mass number of 44 and a latent heat of sublimation of $25200 \mathrm{~J} /$ mole at 194.6 K . In the solid state there are six degrees of freedom per molecule, but there are only five in the vapor state. The average work done per molecule in subliming is $p v=k T$, where $v$ is the volume per molecule.

Lottie Adams

Numerade Educator

01:21

Problem 49

Suppose that the specific heat at constant pressure for ice in units of $\mathrm{J} /(\mathrm{g} \mathrm{K})$ were given by $0.2613 \sqrt{T}$ from 0 K to $64 \mathrm{~K}, 2.09$ from 64 K to 273 K , and 4.18 from 273 K to 293 K . The latent heat of fusion of ice at 273 K is 333 $\mathrm{J} / \mathrm{g}$. Use this information to calculate the entropy of a liter of water at $20^{\circ} \mathrm{C}$.

Manne Andergronde

Numerade Educator