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Heat and Thermodynamics

M. W. Zemansky, Richard H. Dittman

Chapter 4

Heat and the First Law of Thermodynamics - all with Video Answers

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Chapter Questions

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Problem 1

A gas contained in a cylinder by a layer of styrofoam is quickly compressed, the temperature rising several hundred degrees. Has there been a transfer of heat? Has the "heat content" of the gas been increased?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 2

A combustion cxperiment is performed by burning a mixture of fuel and oxygen in a constant-volume container surrounded by a water bath. During the experiment, the temperature of the water rises. If the system is the mixture of fuel and oxygen:
(a) Has heat been transferred?
(b) Has work been done?
(c) What is the sign of $\Delta U$ ?

Ankur S
Ankur S
Numerade Educator
02:18

Problem 3

A liquid is irregularly stirred in a well-insulated container and thereby experiences a rise in temperature. If the system is the liquid:
(a) Has heat been transferred?
(b) Has work been done?
(c) What is the sign of $\Delta U$ ?

Supratim Pal
Supratim Pal
Numerade Educator
01:01

Problem 4

The amount of water in a lake may be increased by action of underground springs, by inflow from a river, and by rain. It may be decreased by various outflows and by cvaporation.
(a) Comment on the question: How much rain is there in the lake?
(b) Comment on the question: How much water in the lake is due to rain?
(c) What concept is analogous to "rain in the lake"?

Narayan Hari
Narayan Hari
Numerade Educator
01:11

Problem 5

A container with rigid well-insulated walls is divided into two parts by a partition. One part contains a gas, and the other is evacuated. If the partition suddenly breaks, show that the initial and final internal energies of the gas are equal. (Note: this process is called an adiabatic free expansion.)

Supratim Pal
Supratim Pal
Numerade Educator
05:23

Problem 6

When an electric current is maintained in an electrolytic cell of slightly acidic water and $1 \mathrm{~mol}$ of water is electrolyzed into hydrogen and oxygen, $2 F$ (faradays) of charge are transferred through a source of emf $\mathcal{E}(1 F \approx 96,500 \mathrm{C} / \mathrm{mol})$. The energy change of the system is $+286,500 \mathrm{~J}$, and $50,000 \mathrm{~J}$ of heat is absorbed. What is $\mathcal{C}$ ?

Angela Deane
Angela Deane
Numerade Educator
03:45

Problem 7

A cylinder with rigid well-insulated walls is divided into two parts by a rigid insulating wall with a small hole in it. A frictionless, insulated piston is held against the perforated partition, thus preventing the gas that is on the other side from seeping through the hole. The gas is maintained at a pressure $P_{i}$ by another frictionless insulated piston. Imagine both pistons to move simultaneously in such a way that, as the gas streams through the hole, the pressure remains at a constant value $P_{i}$ on one side of the dividing wall and at a constant lower value $P_{f}$ on the other side, until all the gas is forced through the hole. (Note: this process is called a throttling process.) Prove that
$$
U_{i}+P_{i} V_{i}=U_{f}+P_{f} V_{f}
$$

Supratim Pal
Supratim Pal
Numerade Educator
02:49

Problem 8

A container of volume $V$ contains $n$ moles of gas at high pressure. Connected to the container is a capillary tube through which the gas may leak slowly out to the atmosphere, where the pressure is $P_{0}$. Surrounding the container and capillary is a water bath, in which is immersed an electrical resistor. The gas is allowed to leak slowly through the capillary into the atmosphere while electrical energy is dissipated in the resistor at such a rate that the temperature of the gas, the container, the capillary, and the water is kept equal to that of the outside air. Show that, after as much gas as possible has leaked out during time interval $t$, the change in internal energy is
$$
\Delta U=\mathcal{E} I t-P_{0}\left(n v_{0}-V\right)
$$
where $v_{0}$ is the molar volume of the gas at atmospheric pressure, $\mathcal{S}$ is the potential difference across the resistor, and $I$ is the current in the resistor.

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
03:18

Problem 9

A thick-walled insulated metal chamber contains $n_{i}$ moles of helium at high pressure
$P_{i}$. It is connected through a valve with a large, almost empty gasholder in which the pressure is maintained at a constant value $P^{\prime}$, very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the gasholder until the pressure on the two sides of the valve is equalized. Prove that
$$
\frac{n_{f}}{n_{i}}=\frac{h^{\prime}-u_{i}}{h^{\prime}-u_{f}}
$$
where $n_{f}=$ number of moles of helium left in the chamber, $u_{i}=$ initial molar internal energy of helium in the chamber, $u_{f}=$ final molar internal energy of helium in the chamber, and $h^{\prime}=u^{\prime}+P^{\prime} v$ (where $u^{\prime}=$ molar internal energy of helium in the gasholder; $v^{\prime}=$ molar volume of helium in the gasholder).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:45

Problem 10

Regarding the internal energy of a hydrostatic system to be a function of $T$ and $P$, derive the following equations:
$(a)$
$\mathrm{d} Q\left[\left(\frac{\partial U}{\partial T}\right)_{P}+P\left(\frac{\partial V}{\partial T}\right)_{P}\right] d T+\left[\left(\frac{\partial U}{\partial P}\right)_{T}+P\left(\frac{\partial V}{\partial P}\right)_{T}\right] d P .$
(b) $\quad\left(\frac{\partial U}{\partial T}\right)_{P}=C_{P}-P V \beta$.
(c) $\quad\left(\frac{\partial U}{\partial P}\right)_{T}=P V \kappa-\left(C_{P}-C_{V}\right) \frac{\kappa}{\beta}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:39

Problem 11

Taking $U$ to be a function of $P$ and $V$, derive the following equations:
(a) $\mathrm{d} Q=\left(\frac{\partial V}{\partial P}\right)_{V} d P+\left[\left(\frac{\partial U}{\partial V}\right)_{P}+P\right] d V$
(b) $\left(\frac{\partial U}{\partial P}\right)_{V}=\frac{C_{V} \kappa}{\beta}$
(c) $\quad\left(\frac{\partial U}{\partial V}\right)_{P}=\frac{C_{P}}{V \beta}-P .$

John Nicolle
John Nicolle
Numerade Educator
03:21

Problem 12

Derive the equations listed in the accompanying table.

Shahab Ullah
Shahab Ullah
Numerade Educator
01:47

Problem 13

Consider using the apparatus shown in Fig. $4-1(a)$, known as the Joule paddle wheel, to determine the specific heat at constant atmospheric pressure. The paddle wheel is driven by a slowly falling weight, and both have a temperature of $14.5^{\circ} \mathrm{C}$. As a result of the work done by the $0.427 \mathrm{~kg}$ mass that falls $1.00 \mathrm{~m}$, the temperature of $1 \mathrm{~kg}$ of water rises $1^{\circ} \mathrm{C}$. Calculate $c_{p}$.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:31

Problem 14

One mole of a gas obeys the van der Waals equation of state:
$$
\left(P+\frac{a}{v^{2}}\right)(v-b)=R T
$$
and its molar internal energy is given by
$$
u=c T-\frac{a}{v},
$$
where $a, b, c$, and $R$ are constants. Calculate the molar heat capacities $c_{V}$ and $c_{P}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:43

Problem 15

The equation of state for a monatomic solid is
$$
P v+f(v)=\Gamma u
$$
where $v$ is the molar volume, $\Gamma$ is the Grüneisen constant, and $u$ is the molar internal energy due to lattice vibrations. Prove that
$$
\Gamma=\frac{\beta v}{c y \kappa^{\prime}}
$$
where $\kappa$ is the isothermal compressibility. This equation, known as the Grüneisen relation, plays an important role in solid-state theory.

Penny Riley
Penny Riley
Numerade Educator
02:49

Problem 16

The molar heat capacity at constant pressure $C_{P} / n$ of a gas varies with the temperature according to the equation
$$
\frac{C_{P}}{n}=a+b T-\frac{c}{T^{2}}
$$
where $a, b$, and $c$ are constants. How much heat is transferred during an isobaric process in which $n$ moles of gas experience a temperature rise from $T_{i}$ to $T_{f} ?$

Supratim Pal
Supratim Pal
Numerade Educator
02:35

Problem 17

The molar heat capacity at constant volume of a metal at low temperatures varies with the temperature according to the equation
$$
\frac{C_{V}}{n}=\left(\frac{124.8}{\Theta}\right)^{3} T^{3}+\gamma T
$$
where $\Theta$ is the Debye temperature, $\gamma$ is a constant, and $C_{V} / n$ is measured in units of $\mathrm{mJ} / \mathrm{mol} \cdot \mathrm{K}$. The first term on the left is the contribution attributable to lattice vibrations and the second term is due to the contribution of free electrons. For copper, $\Theta$ is $343 \mathrm{~K}$ and $\gamma$ is $0.688 \mathrm{~mJ} / \mathrm{mol} \cdot \mathrm{K}^{2}$. How much heat per mole is transferred during a process in which the temperature changes from 2 to $3 \mathrm{~K}$ ?

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
02:00

Problem 18

Suppose that heat conduction occurs at a constant rate $d Q / d t$ in a hollow sphere with an inner radius $r_{1}$ at temperature $T_{1}$ and an outer radius $r_{2}$ at temperature $T_{2}$. Show that for constant thermal conductivity $K$, the temperature difference between the two surfaces is given by
$$
T_{1}-T_{2} \frac{d Q / d t}{4 \pi K}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:10

Problem 19

Two thin concentric spherical shells of radius $0.05 \mathrm{~m}$ and $0.15 \mathrm{~m}$, respectively, have their annular cavity filled with charcoal. When energy is supplied at the steady rate of $10.8 \mathrm{~W}$ to a heater at the center, a temperature difference of $50^{\circ} \mathrm{C}$ is set up between the spheres. Find the thermal conductivity of charcoal.

Nick Johnson
Nick Johnson
Numerade Educator
04:21

Problem 20

The air above the surface of a freshwater lake is at a temperature $T_{A}$, while the water is at its freezing point $T_{i}$, where $T_{A}<T_{i}$. After a time $t$ has elapsed, ice of thickness $y$ has formed. Assuming that the heat, which is liberated when the water freezes, flows up through the ice by conduction and then into the air by natural convection, prove that
$$
\frac{y}{h}+\frac{y^{2}}{2 K}=\frac{T_{i}-T_{A}}{\rho L} t
$$
where $h$ is the convection coefficient per unit area and is assumed constant while ice forms, $K$ is the thermal conductivity of ice, $l$ is the latent heat of fusion of ice, and $\rho$ is the density of ice. (Hint: The temperature of the upper surface is variable. Assume that the ice has a thickness $y$ and imagine an infinitesimal thickness $d y$ to form in time dt.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:33

Problem 21

A solid cylindrical copper rod $0.10 \mathrm{~m}$ long has one end maintained at a constant temperature of $20 \mathrm{~K}$. The other end is blackened and exposed to thermal radiation from a body at $300 \mathrm{~K}$, with no energy lost or gained through the sides of the cylinder. When equilibrium is reached, what is the temperature difference between the two ends? (Hint: Refer to Fig. 4.7.)

Nick Johnson
Nick Johnson
Numerade Educator
02:08

Problem 22

A cylindrical metal can, blackened on the outside, $0.10 \mathrm{~m}$ high and $0.05 \mathrm{~m}$ in diameter, contains liquid ${ }^{4} \mathrm{He}$ at its normal boiling point of $4.22 \mathrm{~K}$, at which its heat of vaporization is $20.4 \mathrm{~kJ} / \mathrm{kg}$. Completely surrounding the helium can are walls maintained at the temperature of liquid nitrogen $(77.35 \mathrm{~K})$, and the intervening space is continuously evacuated to a very low pressure. How much helium is lost per hour?

Nick Johnson
Nick Johnson
Numerade Educator
01:50

Problem 23

The operating temperature of a tungsten filament in an incandescent lamp is $2460 \mathrm{~K}$, and its total emissivity is $0.30$. Find the surface area of the filament of a $100-\mathrm{W}$ lamp.

Narayan Hari
Narayan Hari
Numerade Educator
01:43

Problem 24

A copper wire of length $1.317 \mathrm{~m}$ and diameter $3.26 \times 10^{-4} \mathrm{~m}$ is blackened and placed along the axis of an evacuated glass tube. The wire is connected to a battery, a rheostat, an ammeter, and a voltmeter, and the current is increased until, at the moment the wire is about to melt, the ammeter reads $12.8 \mathrm{~A}$ and the voltmeter reads $20.2 \mathrm{~V}$. Assuming that all the energy supplied was radiated and that the radiation from the glass tube is negligible, calculate the melting temperature of copper.

Nick Johnson
Nick Johnson
Numerade Educator
02:41

Problem 25

The solar constant is the incident energy per unit of time on a unit area of a surface placed at right angles to a sunbeam just outside the earth's atmosphere. The value of the solar constant is $1.37 \mathrm{~kW} / \mathrm{m}^{2}$. The area of a sphere with radius $93,000,000$ miles is $2.79 \times 10^{23} \mathrm{~m}^{2}$, and the surface area of the sun is $6.09 \times 10^{18} \mathrm{~m}^{2}$. Assuming that the sun is a blackbody, calculate its surface temperature,

Narayan Hari
Narayan Hari
Numerade Educator
08:12

Problem 26

(a) A small body with temperature $T$ and emissivity $\epsilon$ is placed in a large evacuated cavity with interior walls kept at temperature $T_{W}$. When $T_{W}-T$ is small, show that the rate of heat transfer by radiation is
$$
\frac{\mathrm{d} Q}{d t}=4 T_{W}^{3} A \in \sigma\left(T_{W}-T\right)
$$
(b) If the body remains at constant pressure, show that the time for the temperature of the body to change from $T_{1}$ to $T_{2}$ is given by
$$
t=\frac{C_{P}}{4 T_{W}^{3} A \epsilon \sigma} \ln \frac{T_{W}-T_{1}}{T_{W}-T_{2}}
$$
(c) Two small blackened spheres of identical size, one of copper and the other of aluminum, are suspended by silk threads within a large hole in a block of melting ice. It is found that it takes $10 \mathrm{~min}$ for the temperature of the aluminum to drop from 276 to $274 \mathrm{~K}$, and $14.2$ min for the copper to drop the same interval of temperature. What is the ratio of specific heats of aluminum and copper? (The densities of $\mathrm{Al}$ and $\mathrm{Cu}$ are $2.70 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and $8.96 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ at $25^{\circ} \mathrm{C}$,
respectively.)

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
03:07

Problem 27

A blackened solid copper sphere with radius of $0.02 \mathrm{~m}$ is placed in an evacuated enclosure with walls kept at $100^{\circ} \mathrm{C}$. In what time does its temperature change from 103 to $102^{\circ} \mathrm{C} ?\left(c_{P}=0.395 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K} ; \rho=8.96 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right.$ at $\left.25^{\circ} \mathrm{C} .\right)$

Narayan Hari
Narayan Hari
Numerade Educator
01:57

Problem 28

In the case of a paramagnetic gas:
(a) Derive the equation
$$
\mathrm{d} Q=\left(\frac{\partial U}{\partial T}\right)_{v, m} d T+\left[\left(\frac{\partial U}{\partial V}\right)_{m, T}+P\right] d V+\left[\left(\frac{\partial U}{\partial m}\right)_{T, V}-\mu_{0} \not \mathcal{A}\right] d m .
$$
(b) Derive expressions for $C_{v, m}, C_{V, \mathcal{H}}, C_{P, m}$, and $C_{P, X}$.

Dan Ni
Dan Ni
Numerade Educator