Heat and Thermodynamics

M. W. Zemansky, Richard H. Dittman

Chapter 5

Ideal Gas - all with Video Answers

Educators

Chapter Questions

Problem 1

A stream of air moves with a speed $w$. Assume that a mass $m$ of air is stopped adiabatically by an obstacle.
(a) Prove that the rise in temperature of this mass of air is given by
$$
\Delta T=\frac{w^{2} M}{5 R}
$$
where $M$ is the molar mass of air.
(b) Calculate $\Delta T$ when $w=600$ miles $/ \mathrm{h}$.
(c) Apply the equation in part $(a)$ to a meteor moving through a stationary atmosphere at a speed of 20 miles/s. What would happen?

Penny Riley

Penny Riley

Numerade Educator

Problem 2

A vertical tank of length greater than $0.76 \mathrm{~m}$ has its top end closed by a tightly fitting frictionless piston of negligible weight. The air inside the cylinder is at an absolute pressure of $1 \mathrm{~atm}(1 \mathrm{~atm}=101,325 \mathrm{~Pa}) .$ The piston is depressed by pouring mercury on it slowly, so that the temperature of the air is maintained constant. What is the height of the air column when mercury starts to spill over the top of the cylinder?

Narayan Hari

Numerade Educator

Problem 3

Mercury is poured into the open end of a J-shaped glass tube, which is closed at the short end, trapping air in that end. How much mercury can be poured in before the mercury overflows? Assume air to act like an ideal gas. The long and short arms are $1 \mathrm{~m}$ and $0.5 \mathrm{~m}$ long, respectively, and effects due to the curvature of the bottom may be neglected. Take atmospheric pressure to be $76 \mathrm{~cm} \mathrm{Hg}$.

Narayan Hari

Numerade Educator

Problem 4

A cylindrical cocktail glass $15 \mathrm{~cm}$ high and $35 \mathrm{~cm}^{2}$ in cross section contains water up to the $10-\mathrm{cm}$ mark. A card is placed over the top and held there while the glass is inverted. When the support for the card is removed, what mass of water must leave the glass in order that the rest of the water will remain in the glass, if one neglects the weight of the card? (Caution: Try this over a sink.)

Nick Johnson

Numerade Educator

Problem 5

Two bulbs containing air, one of which has a volume three times the other, are connected by a tube of negligible volume and are initially at the same temperature. To what temperature must the air in the larger bulb be raised in order that the pressure be doubled? Neglect heat conduction through the air in the connecting tube.

Nick Johnson

Numerade Educator

Problem 6

Expand the following equations in the form
$$
P_{v}=R T\left(1+B P+C P^{2}+\cdots\right)
$$
and determine the second virial coefficient $B$ in each case:
(a) $\left(P+\frac{a}{v^{2}}\right)(v-b)=R T \quad$ (van der Waals equation of state).
(b) $\left(P e^{a / R T}\right)(\nu-b)=R T \quad$ (Dieterici equation of state).
(c) $\left(P+\frac{a}{v^{2} T}\right)(\nu-b)=R T \quad$ (Berthelot equation of state).
(d) $\left[P+\frac{a}{(v+c)^{2} T}\right](v-b)=R T \quad$ (Clausius equation of state).
(e) $P v=R T\left(1+\frac{B^{\prime}}{v}+\frac{C^{\prime}}{v^{2}}+\cdots\right) \quad$ (another type of virial expansion).

Nick Johnson

Numerade Educator

Problem 7

An ideal gas is contained in a cylinder equipped with a frictionless, nonlcaking piston of area $A$. When the pressure is atmospheric $P_{0}$, the piston face is a distance $I$ from the closed end. The gas is compressed by moving the piston a distance $x$. Calculate the spring constant $7 / x$ of the gas:
(a) Under isothermal conditions.
(b) Under adiabatic conditions.
(c) In what respect is a gas cushion superior to a steel spring?
(d) Using Eq. $(4.14)$, show that $C_{P}-C_{V}=n R$ for the ideal gas.

Narayan Hari

Numerade Educator

Problem 8

The temperature of an ideal gas in a tube of very small, constant cross-sectional area varies linearly from one end $(x=0)$ to the other end $(x=L)$ according to the equation
$$
T=T_{0}+\frac{T_{L}-T_{0}}{L} x
$$
If the volume of the tube is $V$ and the pressure $P$ is uniform throughout the tube, show that the equation of state for $n$ moles of gas is given by
$$
P V=n R \frac{T_{L}-T_{0}}{\ln \left(T_{L} / T_{0}\right)}
$$
Show that, when $T_{L}=T_{0}=T$, the equation of state reduces to the obvious one, $P V=n R T$

Surendra Kumar

Numerade Educator

Problem 9

Prove that the work done by an ideal gas with constant heat capacities during a quasi-static adiabatic expansion is equal to:
(a) $W=-C_{V}\left(T_{i}-T_{f}\right)$
(b) $W=\frac{P_{f} V_{f}-P_{i} V_{i}}{\gamma-1}$.
(c) $W=\frac{P_{f} V_{f}}{\gamma-1}\left[1-\left(\frac{P_{i}}{P_{f}}\right)^{(\gamma-1) / \gamma}\right]$.

Nick Johnson

Numerade Educator

Problem 10

(a) Show that the heat transferred during an infinitesimal quasi-static process of an ideal gas can be written
$$
\mathrm{d} Q=\frac{C_{V}}{n R} V d P+\frac{C_{P}}{n R} P d V
$$
Applying this equation to an adiabatic process, show that $P V^{\prime}=$ const.
(b) An ideal gas of volume $0.05 \mathrm{ft}^{3}$ and pressure $120 \mathrm{lb} / \mathrm{in}^{2}$ undergoes a quasi-static adiabatic expansion until the pressure drops to $15 \mathrm{lb} / \mathrm{in}^{2}$. Assuming $\gamma$ to remain constant at the value $1.4$, calculate the final volume. Calculate the work.

Nick Johnson

Numerade Educator

Problem 11

(a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming $\gamma$ to be constant:
$$
T V^{\gamma-1}=\text { const. }
$$
(b) At about $0.1 \mathrm{~ms}$ after detonation of a 20 -kiloton nuclear fission bomb, the "fireball" consists of a sphere of gas with a radius of about $40 \mathrm{ft}$ and a uniform temperature of $300,000 \mathrm{~K}$. Making rough assumptions, estimate the radius at a temperature of $3000 \mathrm{~K}$.

Ashok Prajapati

Ashok Prajapati

Numerade Educator

Problem 12

(a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming $\gamma$ to be constant:
$$
\frac{T}{P(\gamma-1) / \gamma}=\text { const. }
$$
(b) Helium $\left(\gamma=\frac{5}{3}\right)$ at $300 \mathrm{~K}$ and $1 \mathrm{~atm}$ pressure is compressed quasi-statically and adiabatically to a pressure of 5 atm. Assuming that the helium behaves like the ideal gas, calculate the final temperature.

Sachin Rao

Numerade Educator

Problem 13

A horizontal, insulated cylinder contains a frictionless nonconducting piston. On each side of the piston is 54 liters of an inert monatomic ideal gas at 1 atm and $273 \mathrm{~K}$. Heat is slowly supplied to the gas on the left side until the piston has compressed the gas on the right side to $7.59 \mathrm{~atm}$.
(a) How much work is done on the gas on the right side?
(b) What is the final temperature of the gas on the right side?
(c) What is the final temperature of the gas on the left side?
(d) How much heat was added to the gas on the left side?

Nick Johnson

Numerade Educator

Problem 14

An evacuated bottle with nonconducting walls is connected through a valve to a large supply of gas, where the pressure is $P_{0}$ and the temperature is $T_{0}$. The valve is opened slightly, and helium flows into the bottle until the pressure inside the bottle is $P_{0}$. Assuming that the helium behaves like an ideal gas with constant heat capacities, show that the final temperature of the helium in the bottle is $\gamma T_{0}$.

Nick Johnson

Numerade Educator

Problem 15

A thick-walled insulated chamber contains $n_{i}$ moles of helium at high pressure $P_{i}$. It is connected through a valve with a large, almost empty container of helium at constant pressure $P_{0}$, very nearly atmospheric. The valve is opened slightly, and
the helium flows slowly and adiabatically into the container until the pressures on the two sides of the valve are equal. Assuming the helium to behave like an ideal gas with constant heat capacities, show that:
(a) The final temperature of the gas in the chamber is
$$
T_{f}=T_{i}\left(\frac{P_{f}}{P_{i}}\right)^{(\gamma-1) / \gamma}
$$
(b) The number of moles left in the chamber is
$$
n_{f}=n_{i}\left(\frac{P_{f}}{P_{i}}\right)^{1 / \gamma} \text { . }
$$
(c) The final temperature of the gas in the container is
$$
T_{f}=\frac{T_{i}}{\gamma} \frac{1-P_{f} / P_{i}}{1-\left(P_{f} / P_{i}\right)^{1 / \gamma}} .
$$

Nick Johnson

Numerade Educator

Problem 16

(a) If $y$ is the height above sea level, show that the decrease of atmospheric pressure due to a rise of $d y$ is given by
$$
\frac{d P}{P}=-\frac{M g}{R T} d y
$$
where $M$ is the molar mass of air, $g$ is the acceleration of gravity, and $T$ is the temperature at the height $y$.
(b) If the decrease of pressure in part ( $a$ ) is due to an adiabatic expansion, show that
$$
\frac{d P}{P}=\frac{\gamma}{\gamma-1} \frac{d T}{T}
$$
(c) From parts ( $a$ ) and (b), using some of the numerical data of Sec. 5.7, calculate $d T / d y$ in kelvin per kilometer.

Surendra Kumar

Numerade Educator

Problem 17

A steel ball of mass $10 \mathrm{~g}$ is placed in the tube of cross-sectional area $1 \mathrm{~cm}^{2}$ in Rüchhardt's apparatus. The tube is connected to a jar of air having a capacity of 5 liters, the pressure of the air being $76 \mathrm{~cm} \mathrm{Hg}$.
(a) What is the period of vibration for the ball?
(b) If the ball is held initially at a position where the air pressure is exactly atmospheric and then allowed to fall, how far will the ball drop before it starts to come up?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 18

Carbon dioxide is contained in Rüchhardt's apparatus, which has a volume of $5270 \mathrm{~cm}^{3}$, A ball of mass $16.65 \mathrm{~g}$, placed in the tube of cross-sectional area $2.01 \mathrm{~cm}^{2}$, vibrates with a period of $0.834 \mathrm{~s}$. What is $\gamma$ when the barometer reads $72.3 \mathrm{~cm} ?$

Mahnoor Khan

Numerade Educator

Problem 19

Mercury is poured into a U-tube open at both ends until the total length of mercury is
$h$.
(a) If the level of mercury on one side of the tube is depressed and the mercury is allowed to oscillate with small amplitude, show that, neglecting friction, the period $\tau_{1}$ is given by
$$
\tau_{1}=2 \pi \sqrt{\frac{h}{2 g}}
$$
(b) One end of the U-tube is now closed so that the length of the entrapped air column is $L$, and again the mercury is caused to oscillate. Assuming friction to be negligible, the air to be ideal, and the changes of volume to be adiabatic, show that the period $\tau_{2}$ is now
$$
\tau_{2}=2 \pi \sqrt{\frac{h}{2 g+\gamma h_{0} g / L}}
$$
where $h_{0}$ is the height of the barometric column.
(c) Show that
$$
\gamma=\frac{2 L}{h_{0}}\left(\frac{\tau_{1}^{2}}{\tau_{2}^{2}}-1\right)
$$

Emily Anderson

Numerade Educator

Problem 20

Prove that the expression for the speed of a longitudinal wave in an ideal gas may be written
$$
w=\sqrt{\left(\frac{\partial P}{\partial \rho}\right)_{S}}
$$

Chai Santi

Numerade Educator

Problem 21

What is the speed of a longitudinal wave in argon at $293 \mathrm{~K} ?$

Michael Mackenzie

Michael Mackenzie

Numerade Educator

Problem 22

A standing wave of frequency $1100 \mathrm{~Hz}$ in a column of methane at $293 \mathrm{~K}$ produces nodes that are $20 \mathrm{~cm}$ apart. What is $\gamma ?$

Dading Chen

Numerade Educator

Problem 23

The speed of a longitudinal wave in a mixture of helium and neon at $300 \mathrm{~K}$ was found to be $758 \mathrm{~m} / \mathrm{s}$. What is the composition of the mixture?

Matthew Bamidele

Matthew Bamidele

Numerade Educator

Problem 24

The molar mass of iodine is $127 \mathrm{~g}$. A standing wave in iodine vapor at $400 \mathrm{~K}$ produces nodes that are $6.77 \mathrm{~cm}$ apart when the frequency is $1000 \mathrm{~Hz}$. Is iodine vapor monatomic or diatomic?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 25

An open glass tube of uniform cross-section is bent into the shape of an $\mathrm{L}$. One arm is immersed in a liquid of density $\rho$, and the other arm of length $l$ remains in the air in a horizontal position. The tube is rotated with constant angular speed $\omega$ about the axis of the vertical arm. Prove that the height $y$ to which the liquid rises in the vertical arm is equal to
$$
y=\frac{P_{0}\left(1-e^{-\omega^{2} L^{2} M / 2 R T}\right)}{g \rho}
$$
where $P_{0}$ is atmospheric pressure, $M$ is the molar mass of air, and $g$ is the acceleration of gravity.

Hast Aggarwal

Numerade Educator

Problem 26

One mole of an ideal paramagnetic gas obeys Curie's law, with a Curie constant $C_{\mathrm{C}}$. Assume that the internal energy $U$ is a function of $T$ only, so that $d U=C_{V, m} d T$, where $C_{V, m}$ is a constant heat capacity.
(a) Show that the equation of the family of adiabatic surfaces is
$$
\frac{C_{V, m}}{n R} \ln T+\ln V=\frac{\mu_{0} m^{2}}{2 n R C_{\mathrm{C}}}+\ln A
$$
where $A$ is a constant for one surface.
(b) Sketch one of these surfaces on a $T V$ diagram.

Narayan Hari

Numerade Educator

Problem 27

The definition of the average speed of a particle in an ideal gas is
$$
\langle w\rangle=\frac{\sum w_{j}}{N}
$$
Prove that the number of particles striking a unit area of the wall of the container in unit time is equal to
$$
\frac{N(w)}{4 V}
$$

Akshaya Rs

Numerade Educator

Problem 28

The root-mean-square speed $w_{\mathrm{rms}}$ is defined as $\sqrt{\langle w\rangle^{2}} .$ Show that:
(a) $w_{\mathrm{rms}}=\sqrt{\frac{3 k T}{m}}$.
(b) $w_{\text {rms }}=\sqrt{3 / \gamma}$ times the speed of sound.

James Kiss

Numerade Educator