Solved: System A of three particles and system B of five

Point i in Fig. 20-19 represents the initial state of an ideal gas at temperature T. Taking algebraic ~ signs into account, rank the entropy ~ changes that the gas undergoes as it ~ moves, successively and reversibly, from point i to points a, b, c, and d, greatest first.

n four experiments, blocks A Fig.20-19 Question 1. and B, starting at different initial temperatures, were brought together in an insulating box and allowed to reach a common final temperature. The entropy changes for the blocks in the four experiments had the following values (in joules per kelvin), but not necessarily in the order given. Determine which values for A go with which values for B.

A gas, confined to an insulated cylinder, is compressed adiabatically to half its volume. Does the entropy of the gas increase, decrease, or remain unchanged during this process?

An ideal monatomic gas at initial temperature To (in kelvins) expands from initial volume Vo to volume 2Vo by each of the five processes indicated in the T-V diagram of Fig. 20-20. In which process is the expansion (a) isothermal, (b) isobaric (constant pressure), and (c) adiabatic? Explain your answers. (d) In which processes does the entropy of the gas decrease?

In four experiments, 2.5 mol of hydrogen gas undergoes reversible isothermal expansions, starting from the same volume but at different temperatures. The corresponding p-V plots are shown in Fig. 20-21. Rank the situations according to the change in the entropy of the gas, greatest first.

A box contains 100 atoms in a configuration that has 50 atoms in each half of the box. Suppose that you could count the different microstates associated with this configuration at the rate of 100 billion states per second, using a supercomputer. Without written calculation, guess how much computing time you would need: a day, a year, or much more than a year.

Does the entropy per cycle increase, decrease, or remain the same for (a) a Camot engine, (b) a real engine, and (c) a perfect engine (which is, of course, impossible to build)?

Three Camot engines operate between temperature limits of (a) 400 and 500 K, (b) 500 and 600 K, and (c) 400 and 600 K. Each engine extracts the same amount of energy per cycle from the high-temperature reservoir. Rank the magnitudes of the work done by the engines per cycle, greatest first.

An inventor claims to have invented four engines, each of which operates between constant-temperature reservoirs at 400 and 300 K. Data on each engine, per cycle of operation, are: engine A, QH = 200 J, QL = -175 J, and W = 40 J; engine B, QH = 500 J, QL = -200 J, and W = 400 J; engine C, QH = 600 J, QL = -200 J, and W = 400 J; engine D, QH = 100 J, QL = -90 J, and W = 10 J. Of the first and second laws of thermodynamics, which (if either) does each engine violate?

Does the entropy per cycle increase, decrease, or remain the same for (a) a Camot refrigerator, (b) a real refrigerator, and (c) a perfect refrigerator (which is, of course, impossible to build)?

In an experiment, 200 g of aluminum (with a specific heat of 900 J/kg . K) at 100C is mixed with 50.0 g of water at 20.0C, with the mixture thermally isolated. (a) What is the equilibrium temperature? What are the entropy changes of (b) the aluminum, (c) the water, and (d) the aluminum-water system?

A gas sample undergoes a reversible isothermal expansion. Figure 20-23 gives the change IlS in entropy of the gas versus the final volume Vf of the gas. The scale of the vertical axis is set by IlSs = 64 J/K. How many moles are in the sample?

In the irreversible process of Fig. 20-5, let the initial temperatures of the identical blocks Land R be 305.5 and 294.5 K, respectively, and let 215 J be the energy that must be transferred between the blocks in order to reach equilibrium. For the reversible processes of Fig. 20-6, what is IlS for (a) block L, (b) its reservoir, (c) block R, (d) its reservoir, (e) the two-block system, and (f) the system of the two blocks and the two reservoirs?

(a) For 1.0 mol of a monatomic ideal gas taken through the cycle in Fig. 20-24, where VI = 4.00Vo, what is WI Po Vo as the gas goes from state a to state c along path abc? What is IlEin/po Vo in going (b) from b to c and (c) through one full cycle? What is IlS in going (d) from b to c and (e) through one full cycle?

A mixture of 1773 g of water and 227 g of ice is in an initial equilibrium state at 0.000e. The mixture is then, in a reversible process, brought to a second equilibrium state where the water-ice ratio, by mass, is 1.00: 1.00 at 0.000e. (a) Calculate the entropy change of the system during this process. (The heat of fusion for water is 333 kJ/kg.) (b) The system is then returned to the initial equilibrium state in an irreversible process (say, by using a Bunsen burner). Calculate the entropy change of the system during this process. (c) Are your answers consistent with the second law of thermodynamics?

An 8.0 g ice cube at -10C is put into a Thermos flask containing 100 cm3 of water at 20e. By how much has the entropy of the cube-water system changed when equilibrium is reached? The specific heat of ice is 2220 J/kg . K.

In Fig. 20-25, where V 23 = 3.00Vb n moles of a diatomic ideal gas are taken through the cycle with the molecules rotating but not oscillating. What are (a) P21PJ, (b) P31Pb and (c) T3ITI? For path 1 ~ 2, what are (d) WlnRTJ, (e) QlnRTJ, (f) I1EintfnRTJ, and (g) I1SlnR? For path 2 ~ 3, what are (h) WlnRTj, (i) QlnRTb (j) I1EintfnRTJ, (k) I1SlnR? For path 3 ~ 1, what are (1) WlnRTj, (m) QlnRTj, (n) I1EintfnRTb and (0) I1SlnR?

A 2.0 mol sample of an ideal monatomic gas undergoes the reversible process shown in Fig. 20- 26. The scale of the vertical axis is set by Ts = 400.0 K and the scale of the horizontal axis is set by Ss = 20.0 J/K. (a) How much energy is absorbed as heat by the gas? (b) What is the change in the internal 111 Volume Fig.20-25 Problem 17. g 1:: Ts B cO h V 0.. S v f-< 0 Ss Entropy (IlK) energy of the gas? (c) How much Fig.20-26 Problem 18. work is done by the gas?

Suppose 1.00 mol of a monatomic ideal gas is taken from initial pressure PI and volume VI through two steps: (1) an isothermal expansion to volume 2.00VI and (2) a pressure increase to 2.00pI at constant volume. What is Q/PI VI for (a) step 1 and (b) step 2? What is WlpI VI for (c) step 1 and (d) step 2? For the full process, what are (e) I1EintfPI VI and (f) I1S? The gas is returned to its initial state and again taken to the same final state but now through these two steps: (1) an isothermal compression to pressure 2.00pI and (2) a volume increase to 2.00VI at constant pressure. What is Qlpl VI for (g) step 1 and (h) step 2? What is Wipi VI for (i) step 1 and (j) step 2? For the full process, what are (k) I1Ein/PI VI and (I) I1S?

Expand 1.00 mol of an monatomic gas initially at 5.00 kPa and 600 K from initial volume Vi = 1.00 m3 to final volume Vi = 2.00 m3. At any instant during the expansion, the pressure P and volume Vof the gas are related by P = 5.00 exp[(Vi - V)la], withp in kilopascals, Vi and V in cubic meters, and a = 1.00 m3 What are the final (a) pressure and (b) temperature of the gas? (c) How much work is done by the gas during the expansion? (d) What is I1S for the expansion? (Hint: Use two simple reversible processes to find M.)

Energy can be removed from water as heat at and even below the normal freezing point (O.OC at atmospheric pressure) without causing the water to freeze; the water is then said to be supercooled. Suppose a 1.00 g water drop is supercooled until its temperature is that of the surrounding air, which is at - 5.00C. The drop then suddenly and irreversibly freezes, transferring energy to the air as heat. What is the entropy change for the drop? (Hint: Use a three-step reversible process as if the water were taken through the normal freezing point.) The specific heat of ice is 2220 J/kg . K.

An insulated Thermos contains 130 g of water at 80.0e. You put in a 12.0 g ice cube at OC to form a system of ice + original water. (a) What is the equilibrium temperature of the system? What are the entropy changes of the water that was originally the ice cube (b) as it melts and (c) as it warms to the equilibrium temperature? (d) What is the entropy change of the original water as it cools to the equilibrium temperature? (e) What is the net entropy change of the ice + original water system as it reaches the equilibrium temperature?

A Carnot engine whose low-temperature reservoir is at 1 rc has an efficiency of 40%. By how much should the temperature of the high-temperature reservoir be increased to increase the efficiency to 50%?

A Carnot engine absorbs 52 kJ as heat and exhausts 36 kJ as heat in each cycle. Calculate (a) the engine's efficiency and (b) the work done per cycle in kilojoules.

A Carnot engine has an efficiency of 22.0%. It operates between constant-temperature reservoirs differing in temper.ature by 75.0 Co. What is the temperature of the (a) lower-temperature and (b) higher-temperature reservoir?

In a hypothetical nuclear fusion reactor, the fuel is deuterium gas at a temperature of 7 X 108 K. If this gas could be used to operate a Carnot engine with TL = 100C, what would be the engine's efficiency? Take both temperatures to be exact and report your answer to seven significant figures.

A Carnot engine operates between 235C and 115C, absorbing 6.30 X 104 J per cycle at the higher temperature. (a) What is the efficiency of the engine? (b) How much work per cycle is this engine capable of performing?

In the first stage of a two-stage Carnot engine, energy is absorbed as heat QI at temperature TJ, work WI is done, and energy is expelled as heat Q2 at a lower temperature T2. The second stage absorbs that energy as heat Qz, does work W2, and expels energy as heat Q3 at a still lower temperature T3. Prove that the efficiency of the engine is (TI - T3)ITI.

Figure 20-27 shows a reversible cycle through which 1.00 mol of a monatomic ideal gas is taken. Assume that P = 2po, V = 2Vo, Po = 1.01 X 105 Pa, and Vo = 0.0225 m3 Calculate (a) the work done during the cycle, (b) the energy added as heat during stroke abc, and (c) the efficiency of the cy- Volume cle. (d) What is the efficiency of a Fig. 20-27 Problem 29. Carnot engine operating between the highest and lowest temperatures that occur in the cycle? (e) Is this greater than or less than the efficiency calculated in (c)?

A 500 W Carnot engine operates between constant-temperature reservoirs at 100C and 60.0e. What is the rate at which energy is (a) taken in by the engine as heat and (b) exhausted by the engine as heat?

The efficiency of a particular car engine is 25% when the engine does 8.2 kJ of work per cycle. Assume the process is reversible. What are (a) the energy the engine gains per cycle as heat Qgain from the fuel combustion and (b) the energy the engine loses per cycle as heat Qlost. If a tune-up increases the efficiency to 31 %, what are (c) Qgain and (d) Qlost at the same work value?

A Carnot engine is set up to produce a certain work W per cycle. In each cycle, energy in the form of heat QH is transferred to the working substance of 2 c9 OL-__ -L __ ~L-__ _L __ ~ __ ~ 250 300 TH(K) Fig. 20-28 Problem 32. 350 the engine from the higher-temperature thermal reservoir, which is at an adjustable temperature TH The lower-temperature thermal reservoir is maintained at temperature TL = 250 K. Figure 20-28 gives QH for a range of TH The scale of the vertical axis is set by QHs = 6.0 kJ. If THis set at 550 K, what is QH?

Figure 20-29 shows a reversible cycle through which 1.00 mol of a monatomic ideal gas is taken. Volume Vc = ~ 8.00Vb' Process be is an adiabatic ex- ~ pansion, with Pb = 10.0 atm and Vb = p.. 1.00 X 10-3 m3. For the cycle, find (a) the energy added to the gas as heat, (b) the energy leaving the gas as heat, (c) the net work done by the gas, and (d) the efficiency of the cycle.

An ideal gas (1.0 mol) is the Fig. 20-29 Problem 33. working substance in an engine that operates on the cycle shown in Fig. 20-30. Processes Be and DA are reversible and adiabatic. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is the engine efficiency?

The cycle in Fig. 20-31 represents the operation of a gaso line internal combustion engine. Volume V3 = 4.00Vj Assume the gasoline-air intake mixture is an ideal gas with 'Y = 1.30. What are the ratios (a) TzITj, (b) T31Tb (c) T41Tb (d) P31Pb and (e) P4lpj? (f) What is the engine efficiency?

How much work must be done by a Carnot refrigerator to transfer 1.0 J as heat (a) from a reservoir at 7.0C to one at 27C, (b) from a reservoir at -73C to one at 27C, (c) from a reservoir at -173C to one at 27C, and (d) from a reservoir at -223C to one at 27C?

A heat pump is used to heat a building. The outside temperature is 25.0C, and the temperature inside the building is to be maintained at 22C. The pump's coefficient of performance is 3.8, and the heat pump delivers 7.54 MJ as heat to the building each hour. If the heat pump is a Carnot engine working in reverse, at what rate must work be done to run it?

A heat pump is used to heat a building. The outside temperature is 25.0C, and the temperature inside the building is to be maintained at 22C. The pump's coefficient of performance is 3.8, and the heat pump delivers 7.54 MJ as heat to the building each hour. If the heat pump is a Carnot engine working in reverse, at what rate must work be done to run it?

A Carnot air conditioner takes energy from the thermal energy of a room at 70F and transfers it as heat to the outdoors, which is at 96F, For each joule of electric energy required to operate the air conditioner, how many joules are removed from the room?

To make ice, a freezer that is a reverse Carnot engine extracts 42 kJ as heat at -15C during each cycle, with coefficient of performance 5.7. The room temperature is 30.3C. How much (a) energy per cycle is delivered as heat to the room and (b) work per cycle is required to run the freezer?

An air conditioner operating between 93F and 70F is rated at 4000 Btu/h cooling capacity. Its coefficient of performance is 27% of that of a Carnot refrigerator operating between the same two temperatures. What horsepower is required of the air conditioner motor?

The motor in a refrigerator has a power of 200 W. If the freezing compartment is at 270 K and the outside air is at 300 K, and assuming the efficiency of a Carnot refrigerator, what is the maximum amount of energy that can be extracted as heat from the freezing compartment in 10.0 min?

Figure 20-32 represents a Carnot engine that works between temperatures Tj = 400 K and Tz = 150 K and drives a Carnot refrigerator that works between temperatures T3 = 325 K and T4 = 225 K. What is the ratio Q3IQl?

(a) During each cycle, a Carnot engine absorbs 750 J as heat from a high-temperature reservoir at 360 K, with the low-temperature reservoir at 2S0 K. How much work is done per cycle? (b) The engine is then made to work in reverse to function as a Carnot refrigerator between those same two reservoirs. During each cycle, how much work is required to remove 1200 J as heat from the lowtemperature reservoir?

Construct a table like Table 20-1 for eight molecules.

A box contains N identical gas molecules equally divided between its two halves. For N = 50, what are (a) the multiplicity W of the central configuration, (b) the total number of microstates, and (c) the percentage of the time the system spends in the central configuration? For N = 100, what are (d) Wofthe central configuration, (e) the total number of microstates, and (f) the percentage of the time the system spends in the central configuration? For N = 200, what are (g) Wof the central configuration, (h) the total number of microstates, and (i) the percentage of the time the system spends in the central configuration? (j) Does the time spent in the central configuration increase or decrease with an increase in N?

A box contains N gas molecules. Consider the box to be divided into three equal parts. (a) By extension of Eq. 20-20, write a formula for the multiplicity of any given configuration. (b) Consider two configurations: configuration A with equal numbers of molecules in all three thirds of the box, and configuration B with equal numbers of molecules in each half of the box divided into two equal parts rather than three. What is the ratio WAIWB of the multiplicity of configuration A to that of configuration B? (c) Evaluate WAIWB for N = 100. (Because 100 is not evenly divisible by 3, put 34 molecules into one of the three box parts of configuration A and 33 in each of the other two parts.)

Four particles are in the insulated box of Fig. 20-17. What are (a) the least multiplicity, (b) the greatest multiplicity, (c) the least entropy, and (d) the greatest entropy of the four-particle system?

A cylindrical copper rod of length 1.50 m and radius 2.00 cm is insulated to prevent heat loss through its curved sUliace. One end is attached to a thermal reservoir fixed at 300C; the other is attached to a thermal reservoir fixed at 30.0e. What is the rate at which entropy increases for the rod - reservoirs system?

Suppose 0.550 mol of an ideal gas is isothermally and reversibly expanded in the four situations given below. What is the change in the entropy of the gas for each situation? Situation (a) (b) (c) (d) Temperature (K) 250 350 400 450 Initial volume (cm3) 0.200 0.200 0.300 0.300 Final volume (cm3) O.SOO O.SOO 1.20 1.20

As a sample of nitrogen gas (N2) undergoes a temperature increase at constant volume, the distribution of molecular speeds increases. That is, the probability distribution function P(v) for the molecules spreads to higher speed values, as sugPROBLEMS 559 gested in Fig. 19-5b. One way to report the spread in P(v) is to measure the difference i1v between the most probable speed Vp and the rms speed Vrms ' When P(v) spreads to higher speeds, i1v increases. Assume that the gas is ideal and the N2 molecules rotate but do not oscillate. For 1.5 mol, an initial temperature of 250 K, and a final temperature of 500 K, what are (a) the initial difference i1v;, (b) the final difference i1vf, and (c) the entropy change i1S for the gas?

Suppose 1.0 mol of a monatomic ideal gas initially at 10 Land 300 K is heated at constant volume to 600 K, allowed to expand isothermally to its initial pressure, and finally compressed at constant pressure to its original volume, pressure, and temperature. During the cycle, what are (a) the net energy entering the system (the gas) as heat and (b) the net work done by the gas? (c) What is the efficiency ofthe cycle?

Suppose 1.0 mol of a monatomic ideal gas initially at 10 Land 300 K is heated at constant volume to 600 K, allowed to expand isothermally to its initial pressure, and finally compressed at constant pressure to its original volume, pressure, and temperature. During the cycle, what are (a) the net energy entering the system (the gas) as heat and (b) the net work done by the gas? (c) What is the efficiency ofthe cycle?

What is the entropy change for 3.20 mol of an ideal monatomic gas undergoing a reversible increase in temperature from 3S0 K to 425 K at constant volume?

A 600 g lump of copper at SO.OC is placed in 70.0 g of water at 1O.0C in an insulated container. (See Table lS-3 for specific heats.) (a) What is the equilibrium temperature of the copper-water system? What entropy changes do (b) the copper, (c) the water, and (d) the copper-water system undergo in reaching the equilibrium temperature?

Figure 20-33 gives the force magnitude F versus stretch distance x F(N) for a rubber band, with the scale of the Fs - - -- Faxis set by Fs = 1.50 N and the scale of the x axis set by Xs = 3.50 cm. The temperature is 2.00e. When the rubber band is stretched by x = 1.70 cm, at what rate does the entropy of the rubber band change during a small additional stretch?

The temperature of 1.00 mol of a Xs x (em) Fig. 20-33 Problem 56. monatomic ideal gas is raised reversibly from 300 K to 400 K, with its volume kept constant. What is the entropy change of the gas?

Repeat Problem 57, with the pressure now kept constant.

A 0.600 kg sample of water is initially ice at temperature - 20e. What is the sample's entropy change if its temperature is increased to 40C?

A three-step cycle is undergone by 3.4 mol of an ideal diatomic gas: (1) the temperature of the gas is increased from 200 K to 500 K at constant volume; (2) the gas is then isothermally expanded to its original pressure; (3) the gas is then contracted at constant pressure back to its original volume. Throughout the cycle, the molecules rotate but do not oscillate. What is the efficiency of the cycle?

An inventor has built an engine X and claims that its efficiency ex is greater than the efficiency e of an ideal engine operating between the same two temperatures. Suppose you couple engine X to an ideal refrigerator (Fig. 20-34a) and adjust the cycle of engine X so that the work per cycle it provides equals the work per cycle required by the ideal refrigerator. Treat this combination as a single unit and show that if the inventor's claim were true (if ex > e), the combined unit would act as a perfect refrigerator (Fig. 20- 34b), transferring energy as heat from the low-temperature reservoir to the high-temperature reservoir without the need for work.

Suppose 2.00 mol of a diatomic gas is taken reversibly around the cycle shown in the 350 T-S diagram of Fig. 20-35, where g Sj = 6.00 J/K and S2 = 8.00 J/K. ~ The molecules do not rotate or ~ 300

A three-step cycle is undergone reversibly by 4.00 mol of an ideal gas: (1) an adiabatic expansion that gives the gas 2.00 times its initial volume, (2) a constant-volume process, (3) an isothermal compression back to the initial state of the gas. We do not know whether the gas is monatomic or diatomic; if it is diatomic, we do not know whether the molecules are rotating or oscillating. What are the entropy changes for (a) the cycle, (b) process 1, (c) process 3,and (d) process 2?

(a) A Carnot engine operates between a hot reservoir at 320 K and a cold one at 260 K. If the engine absorbs 500 J as heat per cycle at the hot reservoir, how much work per cycle does it deliver? (b) If the engine working in reverse functions as a refrigerator between the same two reservoirs, how much work per cycle must be supplied to remove 1000 J as heat from the cold reservoir?

A 2.00 mol diatomic gas initially at 300 K undergoes this cycle: It is (1) heated at constant volume to 800 K, (2) then allowed to expand isothermally to its initial pressure, (3) then compressed at constant pressure to its initial state. Assuming the gas molecules neither rotate nor oscillate, find (a) the net energy transferred as heat to the gas, (b) the net work done by the gas, and (c) the efficiency of the cycle.

An ideal refrigerator does 150 J of work to remove 560 J as heat from its cold compartment. (a) What is the refrigerator's coefficient of performance? (b) How much heat per cycle is exhausted to the kitchen?

Suppose that 260 J is conducted from a constant-temperature reservoir at 400 K to one at (a) 100 K, (b) 200 K, ( c) 300 K, and (d) 360 K. What is the net change in entropy ilSnet of the reservoirs in each case? (e) As the temperature difference of the two reservoirs decreases, does ilSnet increase, decrease, or remain the same?

An apparatus that liquefies helium is in a room maintained at 300 K. If the helium in the apparatus is at 4.0 K, what is the minimum ratio QtolQfrom, where Qto is the energy delivered as heat to the room and Qfrom is the energy removed as heat from the helium?

A brass rod is in thermal contact with a constanttemperature reservoir at 130DC at one end and a constant-temperature reservoir at 24.0DC at the other end. (a) Compute the total change in entropy of the rod - reservoirs system when 5030 J of energy is conducted through the rod, from one reservoir to the other. (b) Does the entropy of the rod change?

A 45.0 g block of tungsten at 30.0C and a 25.0 g block of silver at -120DC are placed together in an insulated container. (See Table 18-3 for specific heats.) (a) What is the equilibrium temperature? What entropy changes do (b) the tungsten, (c) the silver, and (d) the tungsten-silver system undergo in reaching the equilibrium temperature?

A box contains N molecules. Consider two configurations: configuration A with an equal division of the molecules between the two halves of the box, and configuration B with 60.0% of the molecules in the left half of the box and 40.0% in the right half. For N = 50, what are (a) the multiplicity VVA of configuration A, (b) the multiplicity VVB of configuration B, and (c) the ratio tElA of the time the system spends in configuration B to the time it spends in configuration A? For N = 100, what are (d) VVA , (e) VVB, and (f) tBIA? For N = 200, what are (g) VVA , (h) VVB, and (i) tBIA? (j) With increasing N, does t increase, decrease, or remain the same?

Calculate the efficiency of a fossil-fuel power plant that consumes 380 metric tons of coal each hour to produce useful work at the rate of 750 MW. The heat of combustion of coal (the heat due to burning it) is 28 MJ/kg.

A Carnot refrigerator extracts 35.0 kJ as heat during each cycle, operating with a coefficient of performance of 4.60. What are (a) the energy per cycle transferred as heat to the room and (b) the work done per cycle?

A Carnot engine whose high-temperature reservoir is at 400 K has an efficiency of 30.0%. By how much should the temperature of the low-temperature reservoir be changed to increase the efficiency to 40.0%?

System A of three particles and system B of five particles are in insulated boxes like that in Fig. 20-17. What is the least multiplicity VV of (a) system A and (b) system B? What is the greatest mUltiplicity VV of (c) A and (d) B? What is the greatest entropy of (e) A and (f) B?