Heat and Thermodynamics

M. W. Zemansky, Richard H. Dittman

Chapter 3

Work - all with Video Answers

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Chapter Questions

Problem 1

A thin-walled metal container of volume $V$ contains a gas at high pressure. Connected to the container is a capillary tube and stopcock. When the stopcock is opened slightly, the gas leaks slowly into a cylinder equipped with a nonleaking. frictionless piston, where the pressure remains constant at the atmospheric value $P_{0}$.
(a) Show that, after as much gas as possible has leaked out, an amount of work
$$
W=-P_{0}\left(V_{0}-V\right)
$$

Penny Riley

Penny Riley

Numerade Educator

Problem 2

(a) Show that the work done by an ideal gas during the quasi-static, isothermal expansion from an initial pressure $P_{i}$ to a final pressure $P_{f}$ is given by
$$
W=n R T \ln \frac{P_{f}}{P_{i}}
$$
(b) Calculate the work done when the pressure of 1 mol of an ideal gas is decreased quasi-statically from 20 to $1 \mathrm{~atm}$, the temperature remaining constant at $20^{\circ} \mathrm{C}$ $(R=8.31 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{deg})$

Narayan Hari

Numerade Educator

Problem 3

An adiabatic chamber with rigid walls consists of two compartments, one containing a gas and the other evacuated; the partition between the two compartments is suddenly removed. Is the work done during an infinitesimal portion of this process (called an adiabatic free expansion) equal to $P d V$ ?

Supratim Pal

Numerade Educator

Problem 4

(a) Calculate the work done upon expansion of 1 mol of gas quasi-statically and isothermally from volume $v_{i}$ to a volume $v_{f}$, when the equation of state is
$$
\left(P+\frac{a}{v^{2}}\right)(v-b)=R T
$$
where $a$ and $b$ are the van der Waals constants.
(b) If $a=1.4 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{4} / \mathrm{mol}$ and $b=3.2 \times 10^{-5} \mathrm{~m}^{3} / \mathrm{mol}$, how much work is done
when the gas expands from a volume of 10 liters to a volume of $22.4$ liters at $20^{\circ} \mathrm{C} ?$

Sam Stansfield

Numerade Educator

Problem 5

During a quasi-static expansion of a gas in an adiabatic container, the pressure at any moment is given by the equation
$$
P Y^{\top}=K
$$
where $\gamma$ and $K$ are constants. Show that the work done in expanding from a state $\left(P_{i}, V_{i}\right)$ to a state $\left(P_{f}, V_{f}\right)$ is
$$
W=-\frac{P_{i} V_{i}-P_{f} V_{f}}{\gamma-1}
$$
If the initial pressure and volume are $10^{6} \mathrm{~Pa}$ and $10^{-3} \mathrm{~m}^{3}$, respectively, and the final values are $2 \times 10^{5} \mathrm{~Pa}$ and $3.16 \times 10^{-3} \mathrm{~m}^{3}$, respectively, how much work is donc on a gas having $\gamma=1.4 ?$

Supratim Pal

Numerade Educator

Problem 6

A stationary vertical cylinder, closed at the top, contains a gas whose volume may be changed with the aid of a heavy, frictionless piston of weight $w$.
(a) How much work is done by the external force in compressing the gas by an amount $d V$ by raising the piston a distance $d y$ ?
(b) If this device is used as part of an engine, what expression is appropriate to calculate the net work delivered to or received from the surroundings?
(c) If this device is used only to produce temperature changes of the gas, what expression for work would be appropriate?

VS

Vivek Singh

Numerade Educator

Problem 7

The pressure on $100 \mathrm{~g}$ of nickel is increased quasi-statically and isothermally from 0 to $500 \mathrm{~atm}$. Assuming the density and isothermal compressibility to remain constant at values of $8.90 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and $6.75 \times 10^{-12} \mathrm{~Pa}^{-1}$, respectively, calculate the work.

Narayan Hari

Numerade Educator

Problem 8

(a) The tension in a wire is increased quasi-statically and isothermally from $7_{i}$ to $7_{f}$. If the length, cross-sectional area, and isothermal Young's modulus of the wire remain practically constant, show that the work done is
$$
W=\frac{L}{2 A Y}\left(\not F_{f}^{2}-\not 7_{i}^{2}\right)
$$
(b) The tension in a copper wire $1 \mathrm{~m}$ long and $0.001 \mathrm{~cm}^{2}$ in area is increased quasistatically and isothermally at $20^{\circ} \mathrm{C}$ from 10 to $100 \mathrm{~N}$. How much work is done if the isothermal Young's modulus at $20^{\circ} \mathrm{C}$ is $1.23 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2} ?$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 9

The equation of state of an ideal elastic substance is
$$
7=K T\left(\frac{L}{L_{0}}-\frac{L_{0}^{2}}{L^{2}}\right)
$$
where $K$ is a constant and $L_{0}$ (the value of $L$ at zero tension) is a function of temperature only. Calculate the work necessary to compress the substance from $L=L_{0}$ to $L=L_{0} / 2$ quasi-statically and isothermally.

Nick Johnson

Numerade Educator

Problem 10

Show that the work required to blow a spherical soap bubble of radius $r$ in an isothermal, quasi-static process in the atmosphere is equal to $8 \pi \gamma r^{2}$.

Penny Riley

Numerade Educator

Problem 11

An electrochemical cell, in which the reaction
$$
\mathrm{Cu}+\mathrm{Hg}_{2} \mathrm{SO}_{4} \rightarrow 2 \mathrm{Hg}+\mathrm{CuSo}_{4}
$$
takes place, is connected to a motor having a back emf only slightly smaller than the $\mathrm{emf}$ of the cell. The emf of this cell is given by Eq. (2.14), with $\mathcal{S}_{20}=0.3497 \mathrm{~V}$, $\alpha=-6.35 \times 10^{-4} \mathrm{~V} / \mathrm{deg}, \beta=-2.4 \times 10^{-6} \mathrm{~V} / \mathrm{deg}^{2}$, and $\gamma=0 .$ If the cell is kept at a
constant temperature of $25^{\circ} \mathrm{C}$ and $0.1 \mathrm{~mol}$ of copper reacts, then how much work is done on the motor?

Henry He

Numerade Educator

Problem 12

A dielectric has an equation of state $P=\chi E V$, where $\chi$ is a function of temperature only. Show that the work done in an isothermal, quasi-static change of state is given by
$$
W=\frac{1}{2 V_{\chi}}\left(\mathcal{O}_{f}^{2}-\boldsymbol{P}_{i}^{2}\right)=\frac{V_{\chi}}{2}\left(E_{f}^{2}-E_{i}^{2}\right)
$$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 13

Prove that the work done during a quasi-static isothermal change of state of a paramagnetic substance obeying Curie's law is given by
$$
W=\frac{\mu_{0} T}{2 C_{\mathrm{C}}}\left(m_{f}^{2}-m_{i}^{2}\right)=\frac{\mu_{0} C_{\mathrm{C}}}{2 T}\left(\mathcal{H}_{f}^{2}-\mathcal{F}_{i}^{2}\right)
$$
where $C_{\mathrm{C}}$ is the Curie constant.

Nick Johnson

Numerade Educator

Problem 14

A volume of $200 \mathrm{~cm}^{3}$ of a paramagnetic substance is maintained at constant temperature. The magnetic field is increased quasi-statically and isothermally from 0 to
$10^{6} \mathrm{~A} / \mathrm{m}$. Assume the Curie law to hold and the Curie constant per unit volume to be $1.885 \mathrm{~K} / \mathrm{m}^{3}$
(a) How much work would have to be done if no material were present?
(b) How much work is done to change the total magnetization of the material when the temperature is $300 \mathrm{~K}$ and when it is $1 \mathrm{~K}$ ?
(c) How much work is done to change the total magnetization by the generator supplying the current?

Aja S

Numerade Educator