Chapter 9, Understanding entropy Video Solutions, Thermodynamics : A Complete Undergraduate Course

Problem 1

Give a careful statement of the relationship between entropy and heat. Is entropy extensive or intensive? What dimensions does it have? Under what circumstances can it make sense to think of entropy 'flowing' from one place to another?

Lottie Adams

Numerade Educator

00:47

Problem 2

Why are the cooling towers of a power station so large?

Salamat Ali

Numerade Educator

05:52

Problem 3

A mug of tea has been left to cool from $90^{\circ} \mathrm{C}$ to $18^{\circ} \mathrm{C}$. If there is $0.2 \mathrm{~kg}$ of tea in the mug, and the tea has specific heat capacity $4200 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~kg}^{-\mathrm{t}}$, show that the entropy of the tea has decreased by $185.7 \mathrm{~J} \mathrm{~K}^{-1}$. Comment on the sign of this result.

Khoobchandra Agrawal

Numerade Educator

00:24

Problem 4

Consider inequality (8.19). Is it possible for $\mathrm{d} S$ and $\mathrm{d} Q$ to have opposite signs? If it is, then give an example.

Coach Rye

Numerade Educator

02:45

Problem 5

$1 \mathrm{~kg}$ of water is warmed from $20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ (a) by placing it in contact with a reservoir at $100^{\circ} \mathrm{C}$, (b) by placing it first in contact with a reservoir at $50^{\circ} \mathrm{C}$ until it reaches that temperature, and then in contact with the reservoir at $100^{\circ} \mathrm{C}$, and (c) by operating a reversible heat engine between it and the reservoir at $100^{\circ} \mathrm{C}$. In each case, what are the entropy changes of (i) the water, (ii) the reservoirs, and (iii) the universe? (Assume the heat capacity of water is independent of temperature.)

Penny Riley

Numerade Educator

01:56

Problem 6

Calculate the change in entropy of $1 \mathrm{~kg}$ of water when it is heated from 10 to $100^{\circ} \mathrm{C}$ and completely vaporized, all at $1 \mathrm{~atm}$ pressure. Check your answer by consulting the chart in Figure $16.7$.
Does the change in entropy imply any irreversibility in the process?

Steven Emmel

University of California - Los Angeles

02:25

Problem 7

Two identical bodies of constant heat capacity $C_{p}$ at temperatures $T_{1}$ and $T_{2}$ respectively are used as reservoirs for a heat engine. If the bodies remain at constant pressure, show that the amount of work obtainable is
$$
W=C_{p}\left(T_{1}+T_{2}-2 T_{f}\right)
$$
where $T_{f}$ is the final temperature attained by both bodies. Show that if the most efficient engine is used, then $T_{f}^{2}=T_{1} T_{2}$.

Penny Riley

Numerade Educator

01:36

Problem 8

A heat engine operates between a tank containing $1000 \mathrm{~m}^{3}$ of water and a river at a constant temperature of $10^{\circ} \mathrm{C}$. If the temperature of the tank. is initially $100^{\circ} \mathrm{C}$, what is the maximum amount of work which the heat ungine can perform? Answer the problem algebraically before you substitute in the numbers. Show that your result can be expressed in the form $W=\Delta U-T_{0} \Delta S_{2}$ and interpret the symbols physically.

Shahab Ullah

Numerade Educator

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Problem 9

Calculate the changes in entropy of the universe as a result of the following processes:
(a) A copper block of mass $400 \mathrm{~g}$ and heat capacity $150 \mathrm{~J} \mathrm{~K}^{-1}$ at $100^{\circ} \mathrm{C}$ is placed in a lake at $10^{\circ} \mathrm{C}$.
(b) The same block, now at $10^{\circ} \mathrm{C}$, is dropped from a height $100 \mathrm{~m}$ into the lake.
(c) Two similar blocks at $100{ }^{\circ} \mathrm{C}$ and $10{ }^{\circ} \mathrm{C}$ are joined together (hint: save time by first realizing what the final temperature must be, given that all the heat lost by one block is received by the other, and then reuse previous calculations).
(d) A capacitor of capacitance $1 \mu \mathrm{F}$ is connected to a battery of e.m.f. $100 \mathrm{~V}$ at $0^{\circ} \mathrm{C}$. (N.B. think carefully about what happens when a capacitor is charged from a battery.)
(e) The same capacitor after being charged to $100 \mathrm{~V}$ is discharged through a resistor at $0^{\circ} \mathrm{C}$.
(f) One mole of gas at $0^{\circ} \mathrm{C}$ is expanded reversibly and isothermally to twice its initial volume.
(g) One mole of gas at $0{ }^{\circ} \mathrm{C}$ is expanded reversibly and adiabatically to twice its initial volume.

Rashmi Sinha

Numerade Educator

01:21

Problem 10

A system consists of two different volumes of ideal gas, either side of a fixed thermally insulating partition. On one side there are 2 moles of gas at temperature $T_{1 i}=500 \mathrm{~K}$. On the other side there are 4 moles of gas at temperature $T_{2 i}=200 \mathrm{~K}$. The molar capacity at constant volume is $C_{V, m}=(3 / 2) R$. (i) If the partition is made thermally conducting, find the final temperature throughout the system when equilibrium is attained,

Penny Riley

Numerade Educator

01:50

Problem 11

In a free expansion (also called Joule expansion), $U$ does not change, and no work is done. However, the entropy must increase because the process is irreversible. Is this a counter-example to the relation $\mathrm{d} U=T \mathrm{~d} S-p \mathrm{~d} V ?$ Explain.

Ajay Singhal

Numerade Educator

01:38

Problem 12

The number of ways of choosing $N_{1}$ items from a collection of $N>N_{1}$ items without respect to ordering is
$$
W=\frac{N !}{N_{1} ! N_{2} !}
$$
where $N_{2}=N-N_{1}$. Calculate $\ln W$, making use of Stirling's approximation $\ln n ! \simeq n \ln n-n$, which is very accurate for large $n .$ Express your result in terms of $N$ and $x \equiv N_{1} / N$, and compare with equation $(9.14)$.

Elizabeth Xu

Numerade Educator

01:45

Problem 13

In view of Section 9.3.2, in what sense, if any, is it the case that the mixing of gases is itself a cause of entropy increase? What is the right physical interpretation of equation (9.14)? Discuss the following two perspectives: (i) Mixing has strictly nothing to do with it. Each gas expands just as it would if the other were not there (not in the detailed dynamics, but in the sense of the overall change in its thermodynamic state). The process amounts to a pair of free expansions that happen to be sharing the same space. The entropy change is strictly, fully, and precisely given by that consideration. It is unequivocally unrelated to the mixing itself. (ii) Considered as a whole, we started with a chamber containing $N_{A}+N_{B}$ particles arranged in a rather special way, with those of type $A$ all in one half of the chamber, and those of type $B$ all in the other. By contrast, at the end any given particle could be in either place, but nothing has happened overall to the volume or temperature of the complete system. Therefore the entropy increase is strictly owing to the increase in the number of ways the particles could be spatially configured inside the system, consistent with the macroscopic properties. In short, it is wholly owing to the mixing.

Khoobchandra Agrawal

Numerade Educator

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Problem 14

Extend the argument of Figure $9.15$ to prove that adiabatic surfaces never cross.

Brooke Smith

Numerade Educator