Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China Physics Coaching Class, Yung-Kuo Lim

Chapter 1 Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

Describe briefly the basic principle of the following instruments for making temperature measurements and state in one sentence the special usefulness of each instrument: constant-volume gas thermometer, thermocouple, thermistor.

Sikandar Baig

Numerade Educator

06:18

Problem 2

Describe briefly three different instruments that can be used for the accurate measurement of temperature and state roughly the temperature range in which they are useful and one important advantage of each instrument. Include at least one instrument that is capable of measuring temperatures down to 1 K .

Sikandar Baig

Numerade Educator

07:20

Problem 3

A bimetallic strip of total thickness $x$ is straight at temperature $T$. What is the radius of curvature of the strip, $R$, when it is heated to temperature $T+\Delta T$ ? The coefficients of linear expansion of the two metals are $\alpha_1$ and $\alpha_2$, respectively, with $\alpha_2>\alpha_1$. You may assume that each metal has thickness $x / 2$, and you may assume that $x \ll R$.

João Gabriel Alencar Caribé

Numerade Educator

02:54

Problem 4

An ideal gas is originally confined to a volume $V_1$ in an insulated container of volume $V_1+V_2$. The remainder of the container is evacuated. The partition is then removed and the gas expands to fill the entire container. If the initial temperature of the gas was $T$, what is the final temperature? Justify your answer.

Shahab Ullah

Numerade Educator

02:31

Problem 5

An insulated chamber is divided into two halves of volumes. The left half contains an ideal gas at temperature $T_0$ and the right half is evacuated. A small hole is opened between the two halves, allowing the gas to flow through, and the system comes to equilibrium. No heat is exchanged with the walls. Find the final temperature of the system.

Paul Gabriel

Numerade Educator

03:45

Problem 6

Define heat capacity $C_v$ and calculate from the first principle the numerical value (in calories $/{ }^{\circ} \mathrm{C}$ ) for a copper penny in your pocket, using your best physical knowledge or estimate of the needed parameters.

Amit Srivastava

Numerade Educator

02:46

Problem 7

Specific heat of granite may be: $0.02,0.2,20,2000 \mathrm{cal} / \mathrm{g} \cdot \mathrm{K}$.

Prashant Bana

Numerade Educator

21:35

Problem 8

The figure below shows an apparatus for the determination of $C_p / C_v$ for a gas, according to the method of Clement and Desormes. A bottle $G$, of reasonable capacity (say a few litres), is fitted with a tap $H$, and a manometer $M$. The difference in pressure between the inside and the outside can thus be determined by observation of the difference $h$ in heights of the two columns in the manometer. The bottle is filled with the gas to be investigated, at a very slight excess pressure over the outside atmospheric pressure. The bottle is left in peace (with the tap closed) until the temperature of the gas in the bottle is the same as the outside temperature in the room. Let the reading of the manometer be $h_i$. The tap $H$ is then opened for a very short time, just sufficient for the internal pressure to become equal to the atmospheric pressure (in which case the manometer reads $h=0$ ). With the tap closed the bottle is left in peace for a while, until the inside temperature has become equal to the outside temperature. Let the final reading of the manometer be $h$. From the values of $h_i$ and $h_f$ it is possible to find $C_p / C_v$. (a) Derive an expression for $C_p / C_v$ in terms of $h_i$ and $h_f$ in the above experiment. (b) Suppose that the gas in question is oxygen. What is your theoretical prediction for $C_p / C_v$ at $20^{\circ} \mathrm{C}$, within the framework of statistical mechanics?
FIGURE CAN'T COPY.

Andrew Eddins

Emory University

01:44

Problem 9

(a) Starting with the first law of thermodynamics and the definitions of $c_p$ and $c_v$, show that
$$
c_p-c_v=\left[p+\left(\frac{\partial U}{\partial V}\right)_T\right]\left(\frac{\partial V}{\partial T}\right)_p
$$
where $c_p$ and $c_v$ are the specific heat capacities per mole at constant pressure and volume, respectively, and $U$ and $V$ are energy and volume of one mole.
(b) Use the above results plus the expression
$$
p+\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V
$$
to find $c_p-c_v$ for a Van der Waals gas
$$
\left(p+\frac{a}{V^2}\right)(V-b)=R T \text {. }
$$

Use that result to show that as $V \rightarrow \infty$ at constant $p$, you obtain the ideal gas result for $c_p-c_v$.

Manik Pulyani

Numerade Educator

03:31

Problem 10

One mole of gas obeys Van der Waals equation of state. If its molar internal energy is given by $u=c T-a / V$ (in which $V$ is the molar volume, $a$ is one of the constants in the equation of state, and $c$ is a constant), calculate the molar heat capacities $C_v$ and $C_p$.

Narayan Hari

Numerade Educator

01:18

Problem 11

A solid object has a density $\rho$, mass $M$, and coefficient of linear expansion $\alpha$. Show that at pressure $p$ the heat capacities $C_p$ and $C_v$ are related by
$$
C_p-C_v=3 \alpha M p / \rho .
$$

Hast Aggarwal

Numerade Educator

04:52

Problem 12

One mole of a monatomic perfect gas initially at temperature $T_0$ expands from volume $V_0$ to $2 V_0$, (a) at constant temperature, (b) at constant pressure.

Calculate the work of expansion and the heat absorbed by the gas in each case.

Vipender Yadav

Numerade Educator

05:02

Problem 13

For a diatomic ideal gas near room temperature, what fraction of the heat supplied is available for external work if the gas is expanded at constant pressure? At constant temperature?

Yaqub Khan

Numerade Educator

00:28

Problem 14

A compressor designed to compress air is used instead to compress helium. It is found that the compressor overheats. Explain this effect, assuming the compression is approximately adiabatic and the starting pressure is the same for both gases.

Dading Chen

Numerade Educator

02:37

Problem 15

Calculate the temperature after adiabatic compression of a gas to 10.0 atmospheres pressure from initial conditions of 1 atmosphere and 300 K (a) for air, (b) for helium (assume the gases are ideal).

Sri Datta Vikas Buchemmavari

Numerade Educator

02:20

Problem 16

(a) For a mole of ideal gas at $t=0^{\circ} \mathrm{C}$, calculate the work $W$ done (in Joules) in an isothermal expansion from $V_0$ to $10 V_0$ in volume.
(b) For an ideal gas initially at $t_{\mathrm{i}}=0^{\circ} \mathrm{C}$, find the final temperature $t_f$ (in ${ }^{\circ} \mathrm{C}$ ) when the volume is expanded to $10 \mathrm{~V}_0$ reversibly and adiabatically.

Averell Hause

Carnegie Mellon University

01:41

Problem 17

(a) How much heat is required to raise the temperature of 1000 grams of nitrogen from $-20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ at constant pressure?
(b) How much has the internal energy of the nitrogen increased?
(c) How much external work was done?
(d) How much heat is required if the volume is kept constant?

Take the specific heat at constant volume $c_v=5 \mathrm{cal} / \mathrm{mole}^{\circ} \mathrm{C}$ and $R=2 \mathrm{cal} / \mathrm{mole} \cdot{ }^{\circ} \mathrm{C}$.

Kratika Bhadauria

Numerade Educator

01:35

Problem 18

10 litres of gas at atmospheric pressure is compressed isothermally to a volume of 1 litre and then allowed to expand adiabatically to 10 litres.
(a) Sketch the processes on a $p V$ diagram for a monatomic gas.
(b) Make a similar sketch for a diatomic gas.
(c) Is a net work done on or by the system?
(d) Is it greater or less for the diatomic gas?

Manish Jain

Numerade Educator

10:56

Problem 19

An ideal gas is contained in a large jar of volume $V_0$. Fitted to the jar is a glass tube of cross-sectional area $A$ in which a metal ball of mass $M$ fits snugly. The equilibrium pressure in the jar is slightly higher than atmospheric pressure $p_0$ because of the weight of the ball. If the ball is displaced slightly from equilibrium it will execute simple harmonic motion (neglecting friction). If the states of the gas represent a quasistatic adiabatic process and $\gamma$ is the ratio of specific heats, find a relation between the oscillation frequency $f$ and the variables of the problem.
FIGURE CAN'T COPY.

Nathan Silvano

Numerade Educator

View

Problem 20

The speed of longitudinal waves of small amplitude in an ideal.gas is
$$
C=\sqrt{\frac{d p}{d \rho}}
$$
where $p$ is the ambient gas pressure and $\rho$ is the corresponding gas density. Obtain expressions for
(a) The speed of sound in a gas for which the compressions and rarefactions are isothermal.
(b) The speed of sound in a gas for which the compressions and rarefactions are adiabatic.

Victor Salazar

Numerade Educator

07:54

Problem 21

Two systems with heat capacities $C_1$ and $C_2$, respectively, interact thermally and come to a common temperature $T_f$. If the initial temperature of system 1 was $T_1$, what was the initial temperature of system 2? You may assume that the total energy of the combined systems remains constant.

Surendra Kumar

Numerade Educator

06:17

Problem 22

A large solenoid coil for a physics experiment is made of a single layer of conductor of cross section $4 \mathrm{~cm} \times 2 \mathrm{~cm}$ with a cooling water hole 2 cm $\times 1 \mathrm{~cm}$ in the conductor. The coil, which consists of 100 turns, has a diameter of 3 meters, and a length of 4 meters (the insulation thickness is negligible). At the two ends of the coil are circular steel plates to make the field uniform and to return the magnetic flux through a steel cylindrical structure external to the coil, as shown in the diagram. A magnetic field of 0.25 Tesla is desired. The conductor is made of aluminium.
(a) What power (in kilowatts) must be supplied to provide the desired field, and what must be the voltage of the power supply?
(b) What rate of water flow (litres/second) must be supplied to keep the temperature rise of the water at $40^{\circ} \mathrm{C}$ ? Neglect all heat losses from the coil except through the water.
(c) What is the outward pressure exerted on the coil by the magnetic forces?
(d) If the coil is energized by connecting it to the design voltage calculated in (a), how much time is required to go from zero current to $99 \%$ of the design current? Neglect power supply inductance and resistance. The resistivity of aluminium is $3 \times 10^{-8}$ ohm-meters. Assume that the steel is far below saturation.
FIGURE CAN'T COPY.

Zachary Warner

Numerade Educator

Problem 23

Consider a black sphere of radius $R$ at temperature $T$ which radiates to distant black surroundings at $T=0 \mathrm{~K}$.
(a) Surround the sphere with a nearby heat shield in the form of a black shell whose temperature is determined by radiative equilibrium. What is the temperature of the shell and what is the effect of the shell on the total power radiated to the surroundings?
(b) How is the total power radiated affected by additional heat shields?

Check back soon!

Problem 24

In vacuum insulated cryogenic vessels (Dewars), the major source of heat transferred to the inner container is by radiation through the vacuum jacket. A technique for reducing this is to place "heat shields" in the vacuum space between the inner and outer containers. Idealize this situation by considering two infinite sheets with emissivity $=1$ separated by a vacuum space. The temperatures of the sheets are $T_1$ and $T_2\left(T_2>T_1\right)$. Calculate the energy flux (at equilibrium) between them. Consider a third sheet (the heat shield) placed between the two which has a reflectivity of $R$. Find the equilibrium temperature of this sheet. Calculate the energy flux from sheet 2 to sheet 1 when this heat shield is in place.

For $T_2=$ room temperature, $T_1=$ liquid He temperature ( 4.2 K ) find the temperature of a heat shield that has a reflectivity of $95 \%$. Compare the energy flux with and without this heat shield.
$$
\left(\sigma=0.55 \times 10^{-7} \text { watts } / \mathrm{m}^2 \mathrm{~K}\right)
$$
FIGURE CAN'T COPY.

Check back soon!

Problem 25

Two parallel plates in vacuum, separated by a distance which is small compared with their linear dimensions, are at temperatures $T_1$ and $T_2$ respectively $\left(T_1>T_2\right)$.
(a) If the plates are non-transparent to radiation and have emission powers $e_1$ and $e_2$ respectively, show that the net energy $W$ transferred per unit area per second is
$$
W=\frac{E_1-E_2}{\frac{E_1}{e_1}+\frac{E_2}{e_2}-1} .
$$
where $E_1$ and $E_2$ are the emission powers of black bodies at temperatures $T_1$ and $T_2$ respectively.
(b) Hence, what is $W$ if $T_1$ is 300 K and $T_2$ is 4.2 K , and the plates are black bodies?
(c) What will $W$ be if $n$ identical black body plates are interspersed between the two plates in (b)?
$$
\left(\sigma=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}^4\right)
$$

Check back soon!

04:04

Problem 26

A spherical black body of radius $r$ at absolute temperature $T$ is surrounded by a thin spherical and concentric shell of radius $R$, black on both sides. Show that the factor by which this radiation shield reduces the rate of cooling of the body (consider space between spheres evacuated, with no thermal conduction losses) is given by the following expression: $a R^2 /\left(R^2+b r^2\right)$, and find the numerical coefficients $a$ and $b$.

Surendra Kumar

Numerade Educator

02:12

Problem 27

The solar constant (radiant flux at the surface of the earth) is about $0.1 \mathrm{~W} / \mathrm{cm}^2$. Find the temperature of the sun assuming that it is a black body.

Ajay Singhal

Numerade Educator

08:31

Problem 28

(a) Estimate the temperature of the sun's surface given that the sun subtends an angle $\theta$ as seen from the earth and the earth's surface temperat ure is $T_0$. (Assume the earth's surface temperature is uniform, and that the earth reflects a fraction, $\varepsilon$, of the solar radiation incident upon it). Use your result to obtain a rough estimate of the sun's surface temperature by putting in "reasonable" values for all parameters.
(b) Within an unheated glass house on the earth's surface the temperature is generally greater than $T_0$. Why? What can you say about the maximum possible interior temperature in principle?

Ren Jie Tuieng

Numerade Educator

08:31

Problem 29

Consider an idealized sun and earth, both black bodies, in otherwise empty flat space. The sun is at a temperature of $T_{\mathrm{S}}=6000 \mathrm{~K}$ and heat transfer by oceans and atmosphere on the earth is so effective as to keep the earth's surface temperature uniform. The radius of the earth is $R_{\mathrm{E}}=$ $6 \times 10^8 \mathrm{~cm}$, the radius of the sun is $R_{\mathrm{S}}=7 \times 10^{10} \mathrm{~cm}$, and the earth-sun distance is $d=1.5 \times 10^{13} \mathrm{~cm}$.
(a) Find the temperature of the earth.
(b) Find the radiation force on the earth.
(c) Compare these results with those for an interplanetary "chondrule" in the form of a spherical, perfectly conducting black-body with a radius of $R=0.1 \mathrm{~cm}$, moving in a circular orbit around the sun with a radius equal to the earth-sun distance $d$.

Ren Jie Tuieng

Numerade Educator

04:29

Problem 30

Making reasonable assumptions, estimate the surface temperature of Neptune. Neglect any possible internal sources of heat. What assumptions have you made about the planet's surface and/or atmosphere?

Astronomical data which may be helpful: radius of sun $=7 \times 10^5 \mathrm{~km}$; radius of Neptune $=2.2 \times 10^4 \mathrm{~km}$; mean sun-earth distance $=1.5 \times 10^8 \mathrm{~km}$; mean sun-Neptune distance $=4.5 \times 10^9 \mathrm{~km} ; T_{\mathrm{S}}=6000 \mathrm{~K}$; rate at which sun's radiation reaches earth $=1.4 \mathrm{~kW} / \mathrm{m}^2$; Stefan-Boltzman constant $=$ $5.7 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}^4$.

Jheremiah Simon

Numerade Educator

01:07

Problem 31

A steam turbine is operated with an intake temperature of $400^{\circ} \mathrm{C}$, and an exhaust temperature of $150^{\circ} \mathrm{C}$. What is the maximum amount of work the turbine can do for a given heat input $Q$ ? Under what conditions is the maximum achieved?

Hast Aggarwal

Numerade Educator

04:38

Problem 32

What is a Carnot cycle? Illustrate on a pV diagram and an ST diagram. Derive the efficiency of an engine using the Carnot cycle.

Steven Emmel

University of California - Los Angeles

17:46

Problem 33

A Carnot engine has a cycle pictured below.
FIGURE CAN'T COPY.
(a) What thermodynamic processes are involved at boundaries $A D$ and $B C ; A B$ and $C D$ ?
(b) Where is work put in and where is it extracted?
(c) If the above is a steam engine with $T_{\text {in }}=450 \mathrm{~K}$, operating at room temperature, calculate the efficiency.

Sheh Lit Chang

University of Washington

11:36

Problem 34

A Carnot engine has a cycle as shown in Fig. 1.12. If $W$ and $W^{\prime}$ represent work done by 1 mole of monatomic and diatomic gas, respectively, calculate $W^{\prime} / W$.

Susan Hallstrom

Numerade Educator

02:25

Problem 35

Two identical bodies have internal energy $U=N C T$, with a constant $C$. The values of $N$ and $C$ are the same for each body. The initial temperatures of the bodies are $T_1$ and $T_2$, and they are used as a source of work by connecting them to a Carnot heat engine and bringing them to a common final temperature $T_f$.
(a) What is the final temperature $T_f$ ?
(b) What is the work delivered?

Penny Riley

Numerade Educator

05:56

Problem 36

Water powered machine. A self-contained machine only inputs two equal steady streams of hot and cold water at temperatures $T_1$ and $T_2$. Its only output is a single high-speed jet of water. The heat capacity per unit mass of water, $C$, may be assumed to be independent of temperature. The machine is in a steady state and the kinetic energy in the incoming streams is negligible.
(a) What is the speed of the jet in terms of $T_1, T_2$ and $T$, where $T$ is the temperature of water in the jet?
(b) What is the maximum possible speed of the jet?
FIGURE CAN'T COPY.

Dading Chen

Numerade Educator

02:45

Problem 37

In the water behind a high power dam ( 110 m high) the temperature difference between surface and bottom may be $10^{\circ} \mathrm{C}$. Compare the possible energy extraction from the thermal energy of a gram of water with that generated by allowing the water to flow over the dam through turbines in the conventional way.

Jonathan Everett

Numerade Educator

11:36

Problem 38

Consider an engine working in a reversible cycle and using an ideal gas with constant heat capacity $c_p$ as the working substance. The cycle consists of two processes at constant pressure, joined by two adiabatics.
(a) Find the efficiency of this engine in terms of $p_1, p_2$.
(b) Which temperature of $T_a, T_b, T_c, T_d$ is highest, and which is lowest?
(c) Show that a Carnot engine with the same gas working between the highest and lowest temperatures has greater effficiency than this engine.
FIGURE CAN'T COPY.

Susan Hallstrom

Numerade Educator

04:19

Problem 39

A building at absolute temperature $T$ is heated by means of a heat pump which uses a river at absolute temperature $T_0$ as a source of heat. The heat pump has an ideal performance and consumes power $W$. The building loses heat at a rate $\alpha\left(T-T_0\right)$, where $\alpha$ is a constant.
(a) Show that the equilibrium temperature $T_e$ of the building is given by
$$
T_e=T_0+\frac{W}{2 \alpha}\left[1+\left(1+\frac{4 \alpha T_0}{W}\right)^{\frac{1}{2}}\right] .
$$
(b) Suppose that the heat pump is replaced by a simple heater which also consumes a constant power $W$ and which converts this into heat with $100 \%$ efficiency. Show explicitly why this is less desirable than a heat pump.

Salamat Ali

Numerade Educator

08:38

Problem 40

A room at temperature $T_2$ loses heat to the outside at temperature $T_1$ at a rate $A\left(T_2-T_1\right)$. It is warmed by a heat pump operated as a Carnot cycle between $T_1$ and $T_2$. The power supplied by the heat pump is $d W / d T$.
(a) What is the maximum rate $d Q_m / d t$ at which the heat pump can deliver heat to the room? What is the gain $d Q_m / d W$ ? Evaluate the gain for $t_1=2^{\circ} \mathrm{C}, t_2=27^{\circ} \mathrm{C}$.
(b) Derive an expression for the equilibrium temperature of the room, $T_2$, in terms of $T_1, A$ and $d W / d t$

Vipender Yadav

Numerade Educator

04:19

Problem 41

A building at a temperature $T$ (in K ) is heated by an ideal heat pump which uses the atmosphere at $T_0(\mathrm{~K})$ as heat source. The pump consumes power $W$ and the building loses heat at a rate $\alpha\left(T-T_0\right)$. What is the equilibrium temperature of the building?

Salamat Ali

Numerade Educator

04:37

Problem 42

Let $M$ represent a certain mass of coal which we assume will deliver 100 joules of heat when burned - whether in a house, delivered to the radiators or in a power plant, delivered at $1000^{\circ} \mathrm{C}$. Assume the plant is ideal (no waste in turbines or generators) discharging its heat at $30^{\circ} \mathrm{C}$ to a river. How much heat will $M$, burned at the plant to generate electricity, provide for the house when the electricity is:
(a) delivered to residential resistance-heating radiators?
(b) delivered to a residential heat pump (again assumed ideal) boosting heat from a reservoir at $0^{\circ} \mathrm{C}$ into a hot-air system at $30^{\circ} \mathrm{C}$ ?

Alex Bretton

Numerade Educator

04:15

Problem 43

An air conditioner is a device used to cool the inside of a home. It is, in essence, a refrigerator in which mechanical work is done and heat removed from the (cooler) inside and rejected to the (warmer) outside.

A home air conditioner operating on a reversible Carnot cycle between the inside, absolute temperature $T_2$, and the outside, absolute temperature $T_1>T_2$, consumes $P$ joules $/ \mathrm{sec}$ from the power lines when operating continuously.
(a) In one second, the air conditioner absorbs $Q_2$ joules from the house and rejects $Q_1$ joules outdoors. Develop a formula for the efficiency ratio $Q_2 / P$ in terms of $T_1$ and $T_2$.
(b) Heat leakage into the house follows Newton's law $Q=A\left(T_1-T_2\right)$. Develop a formula for $T_2$ in terms of $T_1, P$, and $A$ for continuous operation of the air conditioner under constant outside temperature $T_1$ and uniform (in space) inside temperature $T_2$.
(c) The air conditioner is controlled by the usual on-off thermostat and it is observed that when the thermostat set at $20^{\circ} \mathrm{C}$ and an outside temperature at $30^{\circ}$, it operates $30 \%$ of the time. Find the highest outside temperature, in ${ }^{\circ} \mathrm{C}$, for which it can maintain $20^{\circ} \mathrm{C}$ inside (use $-273^{\circ} \mathrm{C}$ for absolute zero).
(d) In the winter, the cycle is reversed and the device becomes a heat pump which absorbs heat from outside and rejects heat into the house. Find the lowest outside temperature in ${ }^{\circ} \mathrm{C}$ for which it can maintain $20^{\circ} \mathrm{C}$ inside.
FIGURE CAN'T COPY.

Qiao Ruan

Numerade Educator

02:11

Problem 44

Calculate the change of entropy involved in heating a gram-atomic weight of silver at constant volume from $0^{\circ}$ to $30^{\circ} \mathrm{C}$. The value of $C_v$ over this temperature may be taken as a constant equal to $5.85 \mathrm{cal} / \mathrm{deg}$ mole.

Ajay Singhal

Numerade Educator

15:00

Problem 45

A body of constant heat capacity $C_p$ and a temperature $T_i$ is put into contact with a reservoir at temperature $T_f$. Equilibrium between the body and the reservoir is established at constant pressure. Determine the total entropy change and prove that it is positive for either sign of $\left(T_{\mathrm{f}}-T_{\mathrm{i}}\right) / T_{\mathrm{f}}$. You may regard $\left|T_{\mathrm{f}}-T_{\mathrm{i}}\right| / T_{\mathrm{f}}<1$.

Ravindra Yadav

Numerade Educator

02:45

Problem 46

One kg of $\mathrm{H}_2 \mathrm{O}$ at $0^{\circ} \mathrm{C}$ is brought in contact with a heat reservoir at $100^{\circ} \mathrm{C}$. When the water has reached $100^{\circ} \mathrm{C}$,
(a) what is the change in entropy of the water?
(b) what is the change in entropy of the universe?
(c) how could you heat the water to $100^{\circ} \mathrm{C}$ so the change in entropy of the universe is zero?

Penny Riley

Numerade Educator

03:56

Problem 47

Compute the difference in entropy between 1 gram of nitrogen gas at a temperature of $20^{\circ} \mathrm{C}$ and under a pressure of 1 atm , and 1 gram of liquid nitrogen at a temperature $-196^{\circ} \mathrm{C}$, which is the boiling point of nitrogen, under the same pressure of 1 atm . The latent heat of vaporization of nitrogen is $47.6 \mathrm{cal} / \mathrm{gm}$. Regard nitrogen as an ideal gas with molecular weight 28 , and with a temperature-independent molar specific heat at constant pressure equal to $7.0 \mathrm{cal} / \mathrm{mol} \cdot \mathrm{K}$.

Anthony Pisaturo

Numerade Educator

05:27

Problem 48

A Carnot engine is made to operate as a refrigerator. Explain in detail, with the aid of (a) a pressure-volume diagram, (b) an enthalpy-entropy diagram, all the processes which occur during a complete cycle or operation.

This refrigerator freezes water at $0^{\circ} \mathrm{C}$ and heat from the working substance is discharged into a tank containing water maintained at $20^{\circ} \mathrm{C}$. Determine the minimum amount of work required to freeze 3 kg of water.
FIGURE CAN'T COPY.

Salamat Ali

Numerade Educator

13:10

Problem 49

$n=0.081 \mathrm{kmol}$ of He gas initially at $27^{\circ} \mathrm{C}$ and pressure $=2 \times 10^5 \mathrm{~N} / \mathrm{m}^2$ is taken over the path $A \rightarrow B \rightarrow C$. For He
$$
C_v=3 R / 2, \quad C_p=5 R / 2
$$

Assume the ideal gas law.
(a) How much work does the gas do in expanding at constant pressure from $A \rightarrow B$ ?
(b) What is the change in thermal or internal energy of the helium from $A \rightarrow B$ ?
(c) How much heat is absorbed in going from $A \rightarrow B$ ?
(d) If $B \rightarrow C$ is adiabatic, what is the entropy change and what is the final pressure?

Ceren Uzun

Texas Tech University

02:11

Problem 50

A mole of an ideal gas undergoes a reversible isothermal expansion from volume $V_1$ to $2 V_1$.
(a) What is the change in entropy of the gas?
(b) What is the change in entropy of the universe?

Suppose the same expansion takes place as a free expansion:
(a) What is the change in entropy of the gas?
(b) What is the change in the entropy of the universe?

Manish Kumar ( Iit K )

Numerade Educator

02:59

Problem 51

$N$ atoms of a perfect gas are contained in a cylinder with insulating walls, closed at one end by a piston. The initial volume is $V_1$ and the initial temperature $T_1$.
(a) Find the change in temperature, pressure and entropy that would occur if the volume were suddenly increased to $V_2$ by withdrawing the piston.
(b) How rapidly must the piston be withdrawn for the above expressions to be valid?

Ajay Singhal

Numerade Educator

12:19

Problem 52

A cylinder contains a perfect gas in thermodynamic equilibrium at $p, V, T, U$ (internal energy) and $S$ (entropy). The cylinder is surrounded by a very large heat reservoir at the same temperature $T$. The cylinder walls and piston can be either perfect thermal conductors or perfect thermal insulators. The piston is moved to produce a small volume change $\pm \Delta V$. "Slow" or "fast" means that during the volume change the speed of the piston is very much less than, or very much greater than, molecular speeds at temperature $T$. For each of the five processes below show (on your answer sheet) whether the changes (after the reestablishment of equilibrium) in the other quantities have been positive, negative, or zero.
FIGURE CAN'T COPY.
1. $(+\Delta V)$ (slow) (conduct)
2. $(+\Delta V)$ (slow) (insulate)
3. $(+\Delta V)$ (fast) (insulate)
4. $(+\Delta V)$ (fast) (conduct)
5. $(-\Delta V)$ (fast) (conduct)
TABLE CAN'T COPY.

Keshav Singh

Numerade Educator

02:00

Problem 53

A thermally insulated box is separated into two compartments (volumes $V_1$ and $V_2$ ! by a membrane. One of the compartments contains an ideal gas at temperature $T$; the other is empty (vacuum). The membrane is suddenly removed, and the gas fills up the two comparments and reaches equilibrium.
(a) What is the final temperature of the gas?
(b) Show that the gas expansion process is irreversible.
FIGURE CAN'T COPY.

Ankur S

Numerade Educator

02:25

Problem 54

A thermally conducting, uniform and homogeneous bar of length $L_{\text {, }}$ cross section $A$, density $\rho$ and specific heat at constant pressure $c_p$ is brought to a nonuniform temperature distribution by contact at one end with a hot reservoir at a temperature $T_{\mathrm{H}}$ and at the other end with a cold reservoir at a temperature $T_{\mathrm{c}}$. The bar is removed from the reservoirs, thermally insulated and kept at constant pressure. Show that the change in entropy of the bar is
$$
\Delta S=C_p\left(1+\ln T_{\mathrm{f}}+\frac{T_{\mathrm{c}}}{T_{\mathrm{H}}-T_{\mathrm{c}}} \ln T_{\mathrm{c}}-\frac{T_{\mathrm{H}}}{T_{\mathrm{H}}-T_{\mathrm{c}}} \ln T_{\mathrm{H}}\right),
$$
where $C_p=c_p \rho A L, \quad T_{\mathrm{f}}=\left(T_{\mathrm{H}}+T_{\mathrm{c}}\right) / 2$

Manik Pulyani

Numerade Educator

03:45

Problem 55

A mixture of 0.1 mole of helium $\left(\gamma_1=C_p / C_v=5 / 3\right)$ with 0.2 mole of nitrogen $\left(\gamma_2=7 / 5\right)$, considered an ideal mixture of two ideal gases, is initially at 300 K and occupies 4 litres. Show that the changes of temperature and pressure of the system which occur when the gas is compressed slowly and adiabatically can be described in terms of some intermediate value of $\gamma$. Calculate the magnitude of these changes when the volume is reduced by $1 \%$.

Michael Mackenzie

Numerade Educator

02:05

Problem 56

Consider two ways to mix two perfect gases. In the first, an adiabatically isolated container is divided into two chambers with a pure gas $A$ in the left hand side and a pure gas $B$ in the right. The mixing is accomplished by opening a hole in the dividing wall.
FIGURE CAN'T COPY.
In the second case the chamber is divided by two rigid, perfectly selective membranes, the membrane on the left is perfectly permeable to gas $A$ but impermeable to gas $B$. The membrane on the right is just the reverse. The two membranes are connected by rods to the outside and the whole chamber is connected to a heat reservoir at temperature $T$. The gases can be mixed in this case by pulling left hand membrane to the left and the right hand one to the right.
FIGURE CAN'T COPY.
(a) Find the change in entropy of the container and its contents for second process.
(b) Find the change in entropy of the container and contents for the first process.
(c) What is the change in entropy of the heat reservoir in part (a)?

Penny Riley

Numerade Educator

14:32

Problem 57

Consider a cylinder with a frictionless piston composed of a semipermeable membrane permeable to water only. Let the piston separate a volume $V$ of $N$ moles of pure water from a volume $V^{\prime}$ of a dilute salt $(\mathrm{NaCl})$ solution. There are $N^{\prime}$ moles of water and $n$ moles of the salt in the solution. The system is in contact with a heat reservoir at temperature $T$.
(a) Evaluate an expression for entropy of mixing in the salt solution.
(b) If the piston moves so that the amount of water in the salt solution doubles, how much work is done?
(c) Derive an expression for the pressure $\pi$ across the semipermeable membrane as a function of the volume of the salt solution.

David Morabito

Numerade Educator

02:34

Problem 58

(a) In the big-bang theory of the universe, the radiation energy initially confined in a small region adiabatically expands in a spherically symmetric manner. The radiation cools down as it expands. Derive a relation between the temperature $T$ and the radius $R$ of the spherical volume of radiation, based purely on thermodynamic considerations.
(b) Find the total entropy of a photon gas as a function of its temperature $T$, volume $V$, and the constants $k, \hbar, c$.

Manish Jain

Numerade Educator

15:00

Problem 59

(a) A system, maintained at constant volume, is brought in contact with a thermal reservoir at temperature $T_f$. If the initial temperature of the system is $T_i$, calculate $\Delta S$, change in the total entropy of the system + reservoir. You may assume that $c_v$, the specific heat of the system, is independent of temperature.
(b) Assume now that the change in system temperature is brought about through successive contacts with $N$ reservoirs at temperature $T_1+$ $\Delta T, T_{\mathrm{i}}+2 \Delta T, \ldots, T_{\mathrm{f}}-\Delta T, T_{\mathrm{f}}$, where $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$. Show that in the limit $N \rightarrow \infty, \Delta T \rightarrow 0$ with $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$ fixed, the change in entropy of the system + reservoir is zero.
(c) Comment on the difference between (a) and (b) in the light of the second law of thermodynamics.

Ravindra Yadav

Numerade Educator

04:17

Problem 60

A material is brought from temperature $T_{\mathrm{i}}$ to temperature $T_{\mathrm{i}}$ by placing it in contact with a series of $N$ reservoirs at temperatures $T_{\mathrm{i}}+\Delta T, T_{\mathrm{i}}+$ $2 \Delta T, \ldots, T_{\mathrm{i}}+N \Delta T=T_{\mathrm{f}}$. Assuming that the heat capacity of the material, $C$, is temperature independent, calculate the entropy change of the total system, material plus reservoirs. What is the entropy change in the limit $N \rightarrow \infty$ for fixed $T_1-T_i$ ?

Khoobchandra Agrawal

Numerade Educator

02:45

Problem 61

The specific heat of water is taken as $1 \mathrm{cal} / \mathrm{g} \cdot \mathrm{K}$, independent of temperature, where 1 calorie $=4.18$ joules.
(a) Define the specific heat of a substance at constant pressure in terms of such quantities as $Q$ (heat), $S$ (entropy), and $T$ (temperature).
(b) One kg of water at $0^{\circ} \mathrm{C}$ is brought into sudden contact with a large heat reservoir at $100^{\circ} \mathrm{C}$. When the water has reached $100^{\circ} \mathrm{C}$, what has been the change in entropy of the water? Of the reservoir? Of the entire system consisting of both water and the heat reservoir?
(c) If the water had been heated from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ by first bringing it into contact with a reservoir at $50^{\circ} \mathrm{C}$ and then another reservoir at $100^{\circ} \mathrm{C}$, what would be the change in entropy of the entire system?
(d) Show how the water might be heated from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ with negligible change in entropy of the entire system.

Penny Riley

Numerade Educator

02:25

Problem 62

Two finite, identical, solid bodies of constant total heat capacity per body, $C$, are used as heat sources to drive heat engine. Their initial temperatures are $T_1$ and $T_2$ respectively. Find the maximum work obtainable from the system.

Penny Riley

Numerade Educator

01:31

Problem 63

A rigid box containing one mole of air at temperature $T_0$ (in K ) is initially in thermal contact with an "infinite' heat-capacity reservoir" at the same temperature $T_0$. The box is removed from the reservoir and a cyclic engine is used to take some heat from the reservoir and put some into the air in the box. What is the minimum amount of work from $T_0$ to $T_1$ ? Express $W$ in terms of $T_0, T_1$ and the gas constant $R$, and state units. Ignore vibrational degrees-of-freedom in the air molecules and the heat capacity of the container. Would inclusion of vibrational degrees-offreedom increase or reduce the value of $W$ ?

Manik Pulyani

Numerade Educator

13:13

Problem 64

A reversible heat engine operates between two reservoirs, $T_1$ and $T_2$ $\left(T_2>T_1\right) . T_1$ can be considered to have infinite mass, i.e., $T_1$ remains constant. However the warmer reservoir at $T_2$ consists of a finite amount of gas at constant volume ( $\mu$ moles with a specific heat capacity $C_v$ ).

After the heat engine has operated for some long period of time, the temperature $T_2$ is lowered to $T_1$.
(a) What is the heat extracted from the warmer reservoir during this period?
(b) What is the change of entropy of the warmer reservoir during this period?
(c) How much work did the engine do during this period?

Matthew Muscat

Numerade Educator

06:49

Problem 65

Large heat reservoirs are available at $900 \mathrm{~K}(H)$ and $300 \mathrm{~K}(C)$.
(a) 100 cal of heat are removed from the reservoir $H$ and added to $C$. What is the entropy change of the universe?
(b) A reversible heat engine operates between $H$ and $C$. For each 100 cal of heat removed from $H$, what work is done and what heat is added to $C$ ?
(c) What is the entropy change of the universe in the process of part
(b) above?
(d) A real heat engine is operated as a heat pump removing heat from $C$ and adding heat to $H$. What can be said about the entropy change in the universe produced by the heat pump?

Jonathan Everett

Numerade Educator

02:47

Problem 66

Consider an arbitrary heat engine which operates between two reservoirs, each of which has the same finite temperature-independent heat capacity $c$. The reservoirs have initial temperatures $T_1$ and $T_2$, where $T_2>T_1$, and the engine operates until both reservoirs have the same final temperature $T_3$.
(a) Give the argument which shows that $T_3>\sqrt{T_1 T_2}$.
(b) What is the maximum amount of work obtainable from the engine?

Vipender Yadav

Numerade Educator

09:45

Problem 67

(a) What is the efficiency for a reversible engine operating around the indicated cycle, where $T$ is temperature in K and $S$ is the entropy in joules/K?
FIGURE CAN'T COPY.
(b) A mass $M$ of a liquid at a temperature $T_1$ is mixed with an equal mass of the same liquid at a temperature $T_2$. The system is thermally insulated. If $c_p$ is the specific heat of the liquid, find the total entropy change. Show that the result is always positive.

David Morabito

Numerade Educator

09:43

Problem 68

(a) One mole of an ideal gas is carried from temperature $T_1$ and molar volume $V_1$ to $T_2, V_2$. Show that the change in entropy is
$$
\Delta S=C_v \ln \frac{T_2}{T_1}+R \ln \frac{V_2}{V_1} .
$$
(b) An ideal gas is expanded adiabatically from $\left(p_1, V_1\right)$ to $\left(p_2, V_2\right)$. Then it is compressed isobarically to $\left(p_2, V_1\right)$. Finally the pressure is increased to $p_1$ at constant volume $V_1$. Show that the efficiency of the cycle is
$$
\eta=1-\gamma\left(V_2 / V_1-1\right) /\left(p_1 / p_2-1\right),
$$
where $\gamma=C_p / C_v$.

Averell Hause

Carnegie Mellon University

Problem 69

(1) Suppose you are given the following relation among the entropy $S$, volume $V$, internal energy $U$, and number of particles $N$ of a thermodynamic system: $S=A[N V U]^{1 / 3}$, where $A$ is a constant. Derive a relation among:
(a) $U, N, V$ and $T$;
(b) the pressure $p, N, V$, and $T$.
(c) What is the specific heat at constant volume $c_v$ ?
(2) Now assume two identical bodies each consists solely of a material obeying the equation of state found in part (1). $N$ and $V$ are the same for both, and they are initially at temperatures $T_1$ and $T_2$, respectively. They are to be used as a source of work by bringing them to a common final temperature $T_{\mathrm{f}}$. This process is accomplished by the withdrawal of heat from the hotter body and the insertion of some fraction of this heat in the colder body, the remainder appearing as work.
(a) What is the range of possible final temperatures?
(b) What $T_{\mathrm{f}}$ corresponds to the maximum delivered work, and what is this maximum amount of work?

You may consider both reversible and irreversible processes in answering these questions.

Check back soon!

02:45

Problem 70

One kilogram of water is heated by an electrical resistor from $20^{\circ} \mathrm{C}$ to $99^{\circ} \mathrm{C}$ at constant (atmospheric) pressure. Estimate:
(a) The change in internal energy of the water.
(b) The entropy change of the water.
(c) The factor by which the number of accessible quantum states of the water is increased.
(d) The maximum mechanical work achievable by using this water as heat reservoir to run an engine whose heat sink is at $20^{\circ} \mathrm{C}$.

Penny Riley

Numerade Educator

03:14

Problem 71

One mole of the paramagnetic substance whose TS diagram is shown below is to be used as the working substance in a Carnot refrigerator operating between a sample at 0.2 K and a reservoir at 1 K :
(a) Show a possible Carnot cycle on the TS diagram and describe in detail how the cycle is performed.
(b) For your cycle, how much heat will be removed from the sample per cycle?
(c) How much work will be performed on the paramagnetic substance per cycle?

Konstantin Pavlovskii

Numerade Educator

01:00

Problem 72

A capacitor with a capacity that is temperature sensitive is carried through the following cycle:
(1) The capacitor is kept in a constant temperature bath with a temperature $T_1$ while it is slowly charged (without any ohmic dissipation) to charge $q$ and potential $V_1$. An amount of heat $Q_1$ flows into the capacitor during this charging.
(2) The capacitor is now removed from the bath while charging continues until a potential $V_2$ and temperature $T_2$ are reached.
(3) The capacitor is kept at a temperature $T_2$ and is slowly discharged.
(4) It is removed from the bath which kept it at temperature $T_2$ and discharged completely until it is returned to its initial uncharged state at temperature $T_1$.
(a) Find the net amount of work done in charging and discharging the capacitor.
(b) How much heat flows out of the capacitor in step (3)?
(c) For fixed capacitor charge $q$ find $d V / d T$.
Hint: Consider $V_2=V_1+d V$

Dominador Tan

Numerade Educator

01:30

Problem 73

For each of the following thermodynamic conditions, describe a system, or class of systems (the components or range of components, temperatures, etc.), which satisfies the condition. Confine yourself to classical, single component, chemical systems of constant mass. $U$ is the internal energy and $S$ is the entropy of the system.
(a) $\left(\frac{\partial U}{\partial V}\right)_T=0$,
(b) $\left(\frac{\partial S}{\partial V}\right)_p<0$,
(c) $\left(\frac{\partial T}{\partial S}\right)_p=0$,
(d) $\left(\frac{\partial S}{\partial V}\right)_T=0$,
(e) $\left(\frac{\partial T}{\partial V}\right)_S=-\left(\frac{\partial p}{\partial S}\right)_V$

Manik Pulyani

Numerade Educator

01:45

Problem 74

Consider an ideal gas whose entropy is given by
$$
S=\frac{n}{2}\left[\sigma+5 R \ln \frac{U}{n}+2 R \ln \frac{V}{n}\right],
$$
where $n=$ number of moles, $R=$ universal gas constant, $U=$ internal energy, $V=$ volume, and $\sigma=$ constant.
(a) Calculate $c_p$ and $c_v$, the specific heats at constant pressure and volume.
(b) An old and drafty house is initially in equilibrium with its surroundings at $32^{\circ} \mathrm{F}$. Three hours after turning on the furnace, the house is at a cozy $70^{\circ} \mathrm{F}$. Assuming that the air in the house is described by the above equation, show how the energy density (energy/volume) of the air inside the house compares at the two temperatures.

Khoobchandra Agrawal

Numerade Educator

02:48

Problem 75

A perfect gas may be defined as one whose equation of state is $\mathrm{pV}=$ $N k T$ and whose internal energy is only a function of temperature. For a perfect gas show that
(a) $c_p=c_v+k$, where $c_p$ and $c_v$ are the heat capacities (per molecule) at constant pressure and constant volume respectively.
(b) The quantity $\mathrm{p}^\gamma$ is constant during an adiabatic expansion. (Assume that $\gamma=c_p / c_v$ is constant.)

Supratim Pal

Numerade Educator

05:49

Problem 76

The difference between the speficif heat at constant pressure and the specific heat at constant volume is nearly equal for all simple gases. What is the approximate numerical value of $c_p-c_v$ ? What is the physical reason for the difference between $c_p$ and $c_v$ ? Calculate the difference for an ideal gas.

Shalini Tyagi

Numerade Educator

Problem 77

A paramagnetic system in an uniform magnetic field $H$ is thermally insulated from the surroundings. It has an induced magnetization $M=$ $a H / T$ and a heat capacity $c_H=b / T^2$ at constant $H$, where $a$ and $b$ are constants and $T$ is the temperature. How will the temperature of the system change when $H$ is quasi-statically reduced to zero? In order to have the final temperature change by a factor of 2 from the initial temperature, how strong should be the initial $H$ ?

Check back soon!

05:07

Problem 78

The thermodynamics of a classical paramagnetic system are expressed by the variables: magnetization $M$, magnetic field $B$, and absolute temperature $T$.
The equation of state is
$$
M=C B / T \text {, where } C=\text { Curie constant . }
$$

The system's internal energy is
$$
U=-M B
$$

The increment of work done by the system upon the external environment is $d W=M d B$.
(a) Write an expression for the heat input, $d Q$, to the system in terms of thermodynamic variables $M$ and $B$ :
$$
d Q=(\quad) d M+(\quad) d B .
$$
(b) Find an expression for the differential of the system entropy:
$$
d S=(\quad) d M+(\quad) d B .
$$
(c) Derive an expression for the entropy: $S=$

Carlos Henrique De Lima

Numerade Educator

03:50

Problem 79

The state equation of a new matter is
$$
p=A T^3 / V,
$$
where $p, V$ and $T$ are the pressure, volume and temperature, respectively, $A$ is a constant. The internal energy of the matter is
$$
U=B T^n \ln \left(V / V_0\right)+f(T),
$$
where $B, n$ and $V_0$ are all constants, $f(T)$ only depends on the temperature. Find $B$ and $n$.

Anand Jangid

Numerade Educator

05:09

Problem 80

The following measurements can be made on an elastic band:
(a) The change in temperature when the elastic band is stretched. (In case you have not tried this, hold the attached band with both hands, test the temperature by touching the band to your lips, stretch the band and check the temperature, relax the band and check the temperature once more).
(b) One end of the band is fixed, the other attached to weight $W$, and the frequency $\nu$ of small vibrations is measured.
(c) With the weight at rest $\sigma Q$ is added, and the equilibrium length $L$ is observed to change by $\delta L$.

Derive the equation by which you can predict the result of the last measurement from the results of the first two.

Supratim Pal

Numerade Educator

01:35

Problem 81

The tension $F$ in an ideal elastic cylinder is given by the equation of state
$$
F=a T\left(\frac{L}{L_0(T)}-\frac{L_0^2(T)}{L^2}\right),
$$
where $a$ is a constant, $L_0$ is the length at zero tension, and $L(T)$ is a function of temperature $T$ only.
(a) The cylinder is stretched reversibly and isothermally from $L=L_0$ to $L=2 L_0$. Find the heat transferred to the cylinder, $Q$, in terms of $a, T, L_0$ and $\alpha_0$, the thermal expansion coefficient at zero tension, being
$$
\alpha_0=\frac{1}{L_0(T)} \frac{d L_0(T)}{d T} .
$$
(b) When the length is changed adiabatically, the temperature of the cylinder changes. Derive an expression for the elastocaloric coefficient, $(\partial T / \partial L)_S$ where $S$ is the entropy, in terms of $a, T, L, L_0, \alpha_0$, and $C_L$, the heat capacity at constant length.
(c) Determine whether $C_L$ is a function of $T$ alone, $C_L(T)$, or whether it must also depend on the length, $C_L(T, L)$, for this system.

Nick Johnson

Numerade Educator

02:20

Problem 82

Information: If a rubber band is stretched adiabatically, its temperature increases.
(a) If the rubber band is stretched isothermally, does its entropy increase, decrease, or stay the same?
(b) If the rubber band is stretched adiabatically, does the internal energy increase, decrease, or stay the same?

Qiao Ruan

Numerade Educator

Problem 83

The tension of a rubber band in equilibrium is given by
$$
t=A T\left(\frac{x}{l_0}-\frac{l_0^2}{x^2}\right),
$$
where $t=$ tension, $T=$ absolute temperature, $x=$ length of the band, $l_0$ $=$ length of the band when $t=0, A=$ constant.

When $x$ is the constant length $l_0$, the thermal capacity $c_x(x, T)$ is observed to be a constant $K$.
(a) Find as functions of $T$ and $x$ :
(1) $\left(\frac{\partial E}{\partial x}\right)_T$ where $E=$ internal energy, (2) $\left(\frac{\partial c_x}{\partial x}\right)_T$, (3) $c_x(x, T)$, (4) $E(x, T)$, (5) $S(x, T)$, where $S=$ entropy.
(b) The band is stretched adiabatically from $x=l_0$ to $x=1.5 l_0$. Its initial temperature was $T_0$. What is its final temperature?

Check back soon!

01:43

Problem 84

Consider a gas which undergoes an adiabatic expansion (throttling process) from a region of constant pressure $p_i$ and initial volume $V_{\mathrm{i}}$ to a region with constant pressure $p_{\mathrm{f}}$ and final volume $V_{\mathrm{f}}$ (initial volume 0 ).
FIGURE CAN'T COPY.
(a) By considering the work done by the gas in the process, show that the initial and final enthalpies of the gas are equal.
(b) What can be said about the intermediate states of the system?
(c) Show for small pressure differences $\Delta p=p_t-p_{\mathrm{i}}$ that the temperature difference between the two regions is given by $\Delta T=\frac{V}{c_p}(T \alpha-1) \Delta p$, where $\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$ and $c_p=\left(\frac{\partial U}{\partial T}\right)_p$.
(d) Using the above result, discuss the possibility of using the process to cool either an ideal gas, or a more realistic gas for which $p=R T /(V-b)$. Explain your result.

Penny Riley

Numerade Educator

07:54

Problem 85

(a) Using the equation of state $p V=N R T$ and the specific heat per mole $C_v=3 R / 2$ for a monatomic ideal gas, find its Helmholtz free energy $F$ as a function of number of moles $N, V$, and $T$.
(b) Consider a cylinder separated into two parts by an adiabatic, impermeable piston. Compartments $a$ and $b$ each contains one mole of a monatomic ideal gas, and their initial volumes are $V_{a i}=10$ litres and $V_{b i}=1$ litre, respectively. The cylinder, whose walls allow heat transfer only, is immersed in a large bath at $0^{\circ} \mathrm{C}$. The piston is now moved reversibly so that the final volumes are $V_{a t}=6$ and $V_b=5$ litres. How much work is delivered by (or to) the system?

Khoobchandra Agrawal

Numerade Educator

01:25

Problem 86

A Van der Waal's gas has the equation of state
$$
\left(p+\frac{a}{V^2}\right)(V-b)=R T .
$$
(a) Discuss the physical origin of the parameters $a$ and $b$. Why is the correction to $p$ inversely proportional to $V^2$ ?
(b) The gas undergoes an isothermal expansion from volume $V_1$ to volume $V_2$. Calculate the change in the Helmholtz free energy.
(c) From the information given can you calculate the change in internal energy? Discuss your answer.

Hunza Gilgit

Numerade Educator

02:32

Problem 87

A 100 -ohm resistor is held at a constant temperature of 300 K . A current of 10 amperes is passed through the resistor for 300 sec .
(a) What is the change in the entropy of the resistor?
(b) What is the change in the entropy of the universe?
(c) What is the change in the internal energy of the universe?
(d) What is the change in the Helmholtz free-energy of the universe?

Nicholas Mogoi

Numerade Educator

View

Problem 88

Blackbody radiation.
(a) Derive the Maxwell relation
$$
(\partial S / \partial V)_{\mathrm{T}}=(\partial p / \partial T)_V
$$
(b) From his electromagnetic theory Maxwell found that the pressure $p$ from an isotropic radiation field is equal to $\frac{1}{3}$ the energy density $u(T)$ : $p=\frac{1}{3} u(T)=\frac{U(T)}{3 V}$, where $V$ is the volume of the cavity. Using the first and second laws of thermodynamics together with the result obtained in part (a) show that $u$ obeys the equation
$$
u=\frac{1}{3} T \frac{d u}{d T}-\frac{1}{3} u .
$$
(c) Solve this equation and obtain Stefan's law relating $u$ and $T$.

Andrew Eddins

Emory University

05:07

Problem 89

A magnetic system of spins is at thermodynamic equilibrium at temperature $T$. Let $\mu$ be the magnetic moment of each spin; and let $M$ be the mean magnetization per spin, so $-\mu<M<\mu$. The free energy per spin, for specified magnetization $M$, is $F(M)$.
(1) Compute the magnetization $M$ as a function of external magnetic field strength $B$, given that
$$
F(M)=\lambda \begin{cases}0, & |M / \mu| \leq 1 / 2, \\ (|M / \mu|-1 / 2)^2, & 1 \geq|M / \mu| \geq 1 / 2,\end{cases}
$$
where $\lambda$ is a constant.
(2) Suppose, instead, that someone gives you
$$
F(M)=\lambda\left[(M / \mu)^4-(M / \mu)^2\right],
$$
you should respond that this is unacceptable - this expression violates a fundamental convexity principle of thermodynamics. (a) State the principle. (b) Check it against the above expression. (c) Discuss, by at least one example, what would go wrong with thermodynamics if the principle is not satisfied.

Carlos Henrique De Lima

Numerade Educator

03:31

Problem 90

A certain system is found to have a Gibbs free energy given by
$$
G(p, T)=R T \ln \left[\frac{a p}{(R T)^{5 / 2}}\right]
$$
where $a$ and $R$ are constants. Find the specific heat at constant pressure, $C_p$.

Narayan Hari

Numerade Educator

03:23

Problem 91

Consider a substance held under a pressure $p$ and at a temperature $T$. Show that $(\partial \text { (heat emitted) } / \partial p)_T=T(\partial V / \partial T)_p$

Sheh Lit Chang

University of Washington

02:58

Problem 92

A given type of fuel cell produces electrical energy $W$ by the interaction of $\mathrm{O}_2$ fed into one electrode and $\mathrm{H}_2$ to the other. These gases are fed in at 1 atmosphere pressure and 298 K , and react isothermally and isobarically to form water. Assuming that the reaction occurs reversibly and that the internal resistance of the cell is negligible, calculate the e.m.f. of the cell. Given: one Faraday $=96,500$ coulombs/g mole.

Enthalpies in joules/g mole at 1 atmospheric and 298 K for oxygen, hydrogen, and water are respectively $17,200,8,100$ and $-269,300$.

Entropies in joules/mole- K at 1 atmosphere and 298 K for oxygen, hydrogen, and water are respectively 201, 128 and 66.7 .

Eric Mockensturm

Numerade Educator

Problem 93

It is found for a simple magnetic system that if the temperature $T$ is held constant and the magnetic field $H$ is changed to $H+\Delta H$, the entropy $S$ changes by an amount $\Delta S$,
$$
\Delta S=-\frac{C H \Delta H}{T^2}
$$
where $C$ is a constant characteristic of the system. From this information determine how the magnetization $M$ depends on the temperature and sketch a plot of $M$ versus $T$ for small $H$.

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01:50

Problem 94

A certain magnetic salt is found to obey Curie's law, and to have a heat capacity per unit volume (at constant magnetic field) inversely proportional to the square of the absolute temperature, i.e., $\chi=b / T, c_H=\alpha V / T^2$, where $\alpha=b+a H^2, a$ and $b$ being constants, and $\chi$ is the susceptibility. A sample of this salt at temperature $T_{\mathrm{i}}$ is placed in a magnetic field of strength $H$. The sample is adiabatically demagnetised by slowly reducing the strength of the field to zero. What is the final temperature, $T$, of the salt?

Anand Jangid

Numerade Educator

01:18

Problem 95

Explain the principles of cooling by adiabatic demagnetization. What factors limit the temperature obtained with this method?

Keshav Singh

Numerade Educator

02:34

Problem 96

A flask of conical shape (see figure) contains raw milk. The pressure is measured inside the flask at the bottom. After a sufficiently long time, the cream rises to the top and the milk settles to the bottom. [You may assume that the total volume of liquid remains the same.] Does the pressure increase, decrease, or remain the same? Explain.

Mahendra K

Numerade Educator

06:28

Problem 97

Assume the atmosphere to be an ideal gas of constant specific heat ratio $\gamma=C_p / C_v$. Also assume the acceleration due to gravity, $g$, to be constant over the range of the atmosphere. Let $z=0$ at sea level, $T_0, p_0$, $\rho_0$ be the absolute temperature, pressure, and density of the gas at $z=0$.
(a) Assuming that the thermodynamic variables of the gas are related in the same way they would be for an adiabatic process, find $p(z)$ and $\rho(z)$.
(b) Show that for this case no atmosphere exists above a $z_{\max }$ given by $z_{\max }=\frac{\gamma}{\gamma-1}\left(\frac{R T_0}{g}\right)$, where $R$ is the universal gas constant per gram.

Hariprasad Annamalai

Numerade Educator

15:15

Problem 98

Consider simple models for the earth's atmosphere. Neglect winds, convection, etc, and neglect variation in gravity.
(a) Assume that the atmosphere is isothermal (at $\left.0^{\circ} \mathrm{C}\right)$. Calculate an expression for the distribution of molecules with height. Estimate roughly the height below which half the molecules lie.
(b) Assume that the atmosphere is perfectly adiabatic. Show that the temperature then decreases linearly with height. Estimate this rate of temperature decrease (the so-called adiabatic lapse rate) for the earth.

Averell Hause

Carnegie Mellon University

00:03

Problem 99

The atmosphere is often in a convective steady state at constant entropy, not constant temperature. In such equilibrium $\mathrm{pV}^\gamma$ is independent of altitude, where $\gamma=C_p / C_v$. Use the condition of hydrostatic equilibrium in a uniform gravitational field to find an expression for $d T / d z$, where $z$ is the altitude.

John Johnson

Numerade Educator

15:15

Problem 100

The gas group that is slowly and adiabatically arising and unrestricted near the ground cannot continuously rise; neither can it fall (the atmosphere almost does not convect). If the height $z$ is small, the pressure and temperature of the atmosphere are respectively $p=p_0(1-\alpha z)$ and $T=T_0(1-\beta z)$, where $p_0$ and $T_0$ are respectively the pressure and temperature near the surface. Find $\alpha$ and $\beta$ as functions of the temperature $T_0$, gravitational acceleration near the surface, $g$, and the molecular weight M. Suppose that air consists of $\frac{4}{5} \mathrm{~N}_2$ and $\frac{1}{5} \mathrm{O}_2$, and that $T_0$ is low enough so that the molecule oscillations cannot be excited, but is high enough so that the molecule rotation can be treated by the classical theory.

Averell Hause

Carnegie Mellon University

15:05

Problem 101

Suppose that the earth's atmosphere is an ideal gas with molecular weight $\mu$ and that the gravitational field near the surface is uniform and produces an acceleration $g$.
(a) Show that the pressure $p$ varies as
$$
\frac{1}{p} d p=-\frac{\mu g}{R T} d z
$$
where $z$ is the height above the surface, $T$ is the temperature, and $R$ is the gas constant.
(b) Suppose that the pressure decrease with height is due to adiabatic expansion. Show that
$$
\frac{d p}{p}=\frac{\gamma}{\gamma-1} \frac{d T}{T}, \quad \gamma=\frac{C_p}{C_{\mathrm{v}}} .
$$
(c) Evaluate $d T / d z$ for a pure $N_2$ atmosphere with $\gamma=1.4$.
(d) Suppose the atmosphere is isothermal with temperature $T$. Find $p(z)$ in terms of $T$ and $p_0$, the sea level pressure.
(e) Suppose that at sea level, $p=p_0$ and $T=T_0$. Find $p(z)$ for an adiabatic atmosphere.

Mark Scythian

Numerade Educator

07:38

Problem 102

A fully ionized gas containing a single species of ion with charge $Z|e|$ and atomic weight $A$ is in equilibrium in a uniform gravitational field $g$. The gas is isothermal with temperature $T$ and there is thermal equilibrium between the ions and the electrons. The gas has a low enough density that local interactions between the particles can be neglected.
(a) Show that to avoid charge separation there must be a uniform electric field $E$ given by
$$
E=-\frac{\left(A m_{\mathrm{p}}-m_e\right)}{(1+Z)|e|} g,
$$
where $m_{\mathrm{p}}$ and $m_e$ are the proton and electron masses respectively.
(b) Show that the above equation is also valid if the plasma is not isothermal. (Hint: Treat each component $i$ as an ideal gas subject to the equation of hydrostatic equilibrium
$$
\frac{d p_i}{d x}=n_i F_{i x}
$$
where $p_i$ is the partial pressure of the $i$ th component, $n_i$ is its number density, and $F_{i x}$ is the total force per particle in the $x$ direction.)
(c) The equation in (a) is also valid throughout the sun where $\mathbf{E}$ and g are now directed radially. Show that the charge on the sun is given approximately by
$$
Q=\frac{A}{1+Z} \frac{G M m_p}{|e|},
$$
where $M$ is the mass of the sun.
(d) For the sun $M=2 \times 10^{33}$ grams. If the composition of the sun were pure hydrogen, what would be $Q$ in coulombs? Given this value of $Q$, is the approximation that there is no charge separation a good one?

Urvashi Arora

Numerade Educator

05:13

Problem 103

Consider a thermally isolated system consisting of two volumes, $V$ and 2 V of an ideal gas separated by a thermally conducting and movable partition.
The temperatures and pressures are as shown. The partition is now allowed to move without the gases mixing.

When equilibrium is established what is the change in the total internal energy? The total entropy?
What is the equilibrium temperature? Pressure?
FIGURE CAN'T COPY.

Khoobchandra Agrawal

Numerade Educator

01:56

Problem 104

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder into two parts. Initially, the piston is clamped in the center, with 1 litre of air at 200 K and 2 atm pressure on one side and 1 litre of air at 300 K and 1 atm on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position.
(a) Compute the final pressure and temperature.
(b) Compute the total increase in entropy.

Be sure to give all your reasoning.

Manik Pulyani

Numerade Educator

22:38

Problem 105

A cylindrical container is initially separated by a clamped piston into two compartments of equal volume. The left compartment is filled with one mole of neon gas at a pressure of 4 atmospheres and the right with argon gas at one atmosphere. The gases may be considered as ideal. The whole system is initially at temperature $T=300 \mathrm{~K}$, and is thermally insulated from the outside world. The heat capacity of the cylinder-piston system is $C$ (a constant).
FIGURE CAN'T COPY.
The piston is now unclamped and released to move freely without friction. Eventually, due to slight dissipation, it comes to rest in an equilibrium position. Calculate:
(a) The new temperature of the system (the piston is thermally conductive).
(b) The ratio of final neon to argon volumes.
(c) The total entropy change of the system.
(d) The additional entropy change which would be produced if the piston were removed.
(e) If, in the initial state, the gas in the left compartment were a mole of argon instead of a mole of neon, which, if any, of the answers to (a), (b) and (c) would be different?

Brandy Heflin

Numerade Educator

03:03

Problem 106

Is the melting point of tungsten $350,3500,35,000$, or $350,000^{\circ} \mathrm{C}$ ?

Harsh Gadhiya

Numerade Educator

06:55

Problem 107

Assuming that $1 / 20 \mathrm{eV}$ is required to liberate a molecule from the surface of a certain liquid when $T=300 \mathrm{~K}$, what is the heat of vaporization in ergs/mole?
$$
\left[1 \mathrm{eV}=1.6 \times 10^{-12} \mathrm{erg}\right]
$$

Eduard Sanchez

Numerade Educator

07:35

Problem 108

Twenty grams of ice at $0^{\circ} \mathrm{C}$ are dropped into a beaker containing 120 grams of water initially at $70^{\circ} \mathrm{C}$. Find the final temperature of the mixture neglecting the heat capacity of the beaker. Heat of fusion of ice is $80 \mathrm{cal} / \mathrm{g}$.

Kevin Zaborsky

Numerade Educator

04:24

Problem 109

The entropy of water at atmospheric pressure and $100^{\circ} \mathrm{C}$ is 0.31 $\mathrm{cal} / \mathrm{g} \cdot \mathrm{deg}$, and the entropy of steam at the same temperature and pressure is $1.76 \mathrm{cal} / \mathrm{g} \cdot$ deg.
(a) What is the heat of vaporization at this temperature?
(b) The enthalpy ( $H=U+P V)$ of steam under these conditions is $640 \mathrm{cal} / \mathrm{g}$. Calculate the enthalpy of water under these conditions.

Km Neeraj

Numerade Educator

View

Problem 110

Given 1.0 kg of water at $100^{\circ} \mathrm{C}$ and a very large block of ice at $0^{\circ} \mathrm{C}$. A reversible heat engine absorbs heat from the water and expels heat to the ice until work can no longer be extracted from the system. At the completion of the process:
(a) What is the temperature of the water?
(b) How much ice has been melted? (The heat of fusion of ice is $80 \mathrm{cal} / \mathrm{g})$
(c) How much work has been done by the engine?

Lainey Roebuck

Numerade Educator

05:46

Problem 111

What is the smallest possible time necessary to freeze 2 kg of water at $0^{\circ} \mathrm{C}$ if a 50 watt motor is available and the outside air (hot reservoir) is at $27^{\circ} \mathrm{C}$ ?

Nicholas Mogoi

Numerade Educator

05:47

Problem 112

Compute the theoretical minimum amount of mechanical work needed to freeze 1 kilogram of water, if the water and surroundings are initially at a temperature $T_0=25^{\circ} \mathrm{C}$. The surroundings comprise the only large heat reservoir available.
$$
\left(L_{\text {ice }}=80 \mathrm{cal} / g, \quad C_p=1 \mathrm{cal} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right) .
$$

Matthew Muscat

Numerade Educator

06:32

Problem 113

An ideal Carnot refrigerator (heat pump) freezes ice cubes at the rate of $5 \mathrm{~g} / \mathrm{s}$ starting with water at the freezing point. Energy is given off to the room at $30^{\circ} \mathrm{C}$. If the fusion energy of ice is 320 joules/gram,
(a) At what rate is energy expelled to the room?
(b) At what rate in kilowatts must electrical energy be supplied?
(c) What is the coefficient of performance of this heat pump?

Katie Mcalpine

Numerade Educator