A Carnot heat engine extracts energy Qh from a hot

The only form of energy possessed by molecules of a monatomic ideal gas is translational kinetic energy. Using the results from the discussion of kinetic theory in Section 10.5, show that the internal energy of a monatomic ideal gas at pressure P and occupying volume V may be written as .

Sketch a PV diagram and nd the work done by the gas during the following stages: (a) A gas is expanded from a volume of 1.0 L to 3.0 L at a constant pressure of 3.0 atm. (b) The gas is then cooled at constant volume until the pressure falls to 2.0 atm. (c) The gas is then compressed at a constant pressure of 2.0 atm from a volume of 3.0 L to 1.0 L. (Note: Be careful of signs.) (d) The gas is heated until its pressure increases from 2.0 atm to 3.0 atm at a constant volume. (e) Find the net work done during the complete cycle.

A container of volume 0.40 m3 contains 3.0 mol of argon gas at 30 C. Assuming argon behaves as an ideal gas, nd the total internal energy of the gas. (Hint: See Problem 1.)

A 40.0-g projectile is launched by the expansion of hot gas in an arrangement shown in Figure P12.4a. The crosssectional area of the launch tube is 1.0 cm2, and the length that the projectile travels down the tube after starting from rest is 32 cm. As the gas expands, the pressure varies as shown in Figure P12.4b. The values for the initial pressure and volume are Pi  11  105 Pa and Vi  8.0 cm3 while the nal values are Pf  1.0  105 Pa and Vf  40.0 cm3. Friction between the projectile and the launch tube is negligible. (a) If the projectile is launched into a vacuum, what is the speed of the projectile as it leaves the launch tube? (b) If instead the projectile is launched into air at a pressure of 1.0  105 Pa, what tube is spent by the projectile pushing air out of the way as it proceeds down the tube?

A gas expands from I to F along the three paths indicated in Figure P12.5. Calculate the work done on the gas along paths (a) IAF, (b) IF, and (c) IBF.

Sketch a PV diagram of the following processes: (a) A gas expands at constant pressure P1 from volume V1 to volume V2. It is then kept at constant volume while the pressure is reduced to P2. (b) A gas is reduced in pressure from P1 to P2 while its volume is held constant at V1. It is then expanded at constant pressure P2 to a nal volume V2. (c) In which of the processes is more work done by the gas? Why?

Gas in a container is at a pressure of 1.5 atm and a volume of 4.0 m3. What is the work done on the gas (a) if it expands at constant pressure to twice its initial volume? (b) if it is compressed at constant pressure to one-quarter its initial volume?

A movable piston having a mass of 8.00 kg and a crosssectional area of 5.00 cm2 traps 0.200 moles of an ideal gas in a vertical cylinder. If the piston slides without friction in the cylinder, how much work is done on the gas when its temperature is increased from 20 C to 300 C?

One mole of an ideal gas initially at a temperature of Ti  0 C undergoes an expansion at a constant pressure of 1.00 atm to four times its original volume. (a) Calculate the new temperature Tf of the gas. (b) Calculate the work done on the gas during the expansion.

(a) Determine the work done on a uid that expands from i to f as indicated in Figure P12.10. (b) How much work is done on the uid if it is compressed from f to i along the same path?

A container is placed in a water bath and held at constant volume as a mixture of fuel and oxygen is burned inside it. The temperature of the water is observed to rise during the burning. (The water is also held at constant volume. (a) Consider the burning mixture to be the system. What are the signs of Q, U, and W? (b) What are the signs of these quantities if the water bath is considered to be the system?

A quantity of a monatomic ideal gas undergoes a process in which both its pressure and volume are doubled as shown in Figure P12.12. What is the energy absorbed by heat into the gas during this process? (Hint: See Problem 1.)

A gas is compressed at a constant pressure of 0.800 atm from 9.00 L to 2.00 L. In the process, 400 J of energy leaves the gas by heat. (a) What is the work done on the gas? (b) What is the change in its internal energy?

A monatomic ideal gas undergoes the thermodynamic process shown in the PV diagram of Figure P12.14. Determine whether each of the values U, Q, and W for the gas is positive, negative, or zero. (Hint: See Problem 1.)

A gas expands from I to F in Figure P12.5. The energy added to the gas by heat is 418 J when the gas goes from I to F along the diagonal path. (a) What is the change in internal energy of the gas? (b) How much energy must be added to the gas by heat for the indirect path IAF to give the same change in internal energy?

A gas is taken through the cyclic process described by Figure P12.16. (a) Find the net energy transferred to the system by heat during one complete cycle. (b) If the cycle is reversedthat is, the process follows the path ACBA what is the net energy transferred by heat per cycle?

A gas is enclosed in a container tted with a piston of cross-sectional area 0.150 m2. The pressure of the gas is maintained at 6 000 Pa as the piston moves inward 20.0 cm. (a) Calculate the work done by the gas. (b) If the internal energy of the gas decreases by 8.00 J, nd the amount of heat removed from the system by heat during the compression.

A gas is enclosed in a container tted with a piston of cross-sectional area 0.150 m2. The pressure of the gas is maintained at 6 000 Pa as the piston moves inward 20.0 cm. (a) Calculate the work done by the gas. (b) If the internal energy of the gas decreases by 8.00 J, nd the amount of heat removed from the system by heat during the compression.

One gram of water changes to ice at a constant pressure of 1.00 atm and a constant temperature of 0 C. In the process, the volume changes from 1.00 cm3 to 1.09 cm3. (a) Find the work done on the water and (b) the change in the internal energy of the water.

A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. If at the same time 220 J of work is done on the system, nd the energy transferred to or from it by heat.

A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. If at the same time 220 J of work is done on the system, nd the energy transferred to or from it by heat.

One mole of gas initially at a pressure of 2.00 atm and a volume of 0.300 L has an internal energy equal to 91.0 J. In its nal state, the gas is at a pressure of 1.50 atm and a volume of 0.800 L, and its internal energy equals 180 J. For the paths IAF, IBF, and IF in Figure P12.22, calculate (a) the work done on the gas and (b) the net energy transferred to the gas by heat in the process.

A heat engine operates between two reservoirs at temperatures of 20 C and 300 C. What is the maximum efciency possible for this engine?

A heat engine operates between two reservoirs at temperatures of 20 C and 300 C. What is the maximum efciency possible for this engine?

The energy absorbed by an engine is three times greater than the work it performs. (a) What is its thermal efciency? (b) What fraction of the energy absorbed is expelled to the cold reservoir?

A particular engine has a power output of 5.00 kW and an efciency of 25.0%. If the engine expels 8 000 J of energy in each cycle, nd (a) the energy absorbed in each cycle and (b) the time required to complete each cycle.

One of the most efcient engines ever built is a coal-red steam turbine engine in the Ohio River valley, driving an electric generator as it operates between 1 870 C and 430 C. (a) What is its maximum theoretical efciency? (b) Its actual efciency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40  105 J of energy each second from the hot reservoir?

A gun is a heat engine. In particular, it is an internal combustion piston engine that does not operate in a cycle, but comes apart during its adiabatic expansion process. A certain gun consists of 1.80 kg of iron. It res one 2.40-g bullet at 320 m/s with an energy efciency of 1.10%. Assume that the body of the gun absorbs all of the energy exhaust and increases uniformly in temperature for a short time before it loses any energy by heat into the environment. Find its temperature increase.

An engine absorbs 1 700 J from a hot reservoir and expels 1 200 J to a cold reservoir in each cycle. (a) What is the engines efciency? (b) How much work is done in each cycle? (c) What is the power output of the engine if each cycle lasts for 0.300 s?

A power plant has been proposed that would make use of the temperature gradient in the ocean. The system is to operate between 20.0 C (surface water temperature) and 5.00 C (water temperature at a depth of about 1 km). (a) What is the maximum efciency of such a system? (b) If the useful power output of the plant is 75.0 MW, how much energy is absorbed per hour? (c) In view of your answer to (a), do you think such a system is worthwhile (considering that there is no charge for fuel)?

In one cycle, a heat engine absorbs 500J from a hightemperature reservoir and expels 300J to a low-temperature reservoir. If the efciency of this engine is 60% of the efciency of a Carnot engine, what is the ratio of the low temperature to the high temperature in the Carnot engine?

A heat engine operates in a Carnot cycle between 80.0 C and 350 C. It absorbs 21 000 J of energy per cycle from the hot reservoir. The duration of each cycle is 1.00 s. (a) What is the mechanical power output of this engine? (b) How much energy does it expel in each cycle by heat?

A nuclear power plant has an electrical power output of 1 000 MW and operates with an efciency of 33%. If excess energy is carried away from the plant by a river with a ow rate of 1.0  106 kg/s, what is the rise in temperature of the owing water?

A 20.0%-efcient real engine is used to speed up a train from rest to 5.00 m/s. It is known that an ideal (Carnot) engine using the same cold and hot reservoirs would accelerate the same train from rest to a speed of 6.50 m/s using the same amount of fuel. If the engines use air at 300 K as a cold reservoir, nd the temperature of the steam serving as the hot reservoir.

A freezer is used to freeze 1.0 L of water completely into ice. The water and the freezer remain at a constant temperature of T  0 C. Determine (a) the change in the entropy of the water and (b) the change in the entropy of the freezer.

What is the change in entropy of 1.00 kg of liquid water at 100 C as it changes to steam at 100 C?

A 70-kg log falls from a height of 25 m into a lake. If the log, the lake, and the air are all at 300 K, nd the change in entropy of the Universe during this process.

Two 2 000-kg cars, both traveling at 20 m/s, undergo a head-on collision and stick together. Find the change in entropy of the Universe resulting from the collision if the temperature is 23 C.

The surface of the Sun is approximately at 5 700 K, and the temperature of the Earths surface is approximately 290 K. What entropy change occurs when 1 000 J of energy is transferred by heat from the Sun to the Earth?

Repeat the procedure used to construct Table 12.3 (a) for the case in which you draw three marbles rather than four from your bag and (b) for the case in which you draw ve rather than four

Prepare a table like Table 12.3 for the following occurrence: You toss four coins into the air simultaneously and record all the possible results of the toss in terms of the numbers of heads and tails that can result. (For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved.) (a) On the basis of your table, what is the most probable result of a toss? In terms of entropy, (b) what is the most ordered state, and (c) what is the most disordered?

Consider a standard deck of 52 playing cards that has been thoroughly shufed. (a) What is the probability of drawing the ace of spades in one draw? (b) What is the probability of drawing any ace? (c) What is the probability of drawing any spade?

A student claims that she has constructed a heat engine that operates between the temperatures of 200 K and 100 K with 60% efciency. The professor does not give her credit for the project. Why not?

A Carnot engine operates between the temperatures Th  100 C and Tc  20 C. By what factor does the theoretical efciency increase if the temperature of the hot reservoir is increased to 550 C?

A Carnot heat engine extracts energy Qh from a hot reservoir at constant temperature Th and rejects energy Qc to a cold reservoir at constant temperature Tc. Find the entropy changes of (a) the hot reservoir, (b) the cold reservoir, (c) the engine, and (d) the complete system.

One end of a copper rod is in thermal contact with a hot reservoir at T  500 K, and the other end is in thermal contact with a cooler reservoir at T  300 K. If 8 000 J of energy is transferred from one end of the rod to the other, with no change in the temperature distribution, nd the entropy change of each reservoir and the total entropy change of the Universe.

Find the change in temperature of a river due to the exhausted energy from a nuclear power plant. Assume that the input power to the boiler in the plant is 25  108 W, the efciency of use of this power is 30%, and the river ow rate is 9.0  106 kg/min.

A Carnot engine operates between 100 C and 20 C. How much ice can the engine melt from its exhaust after it has done 5.0  104 J of work?

A 1500-kW heat engine operates at 25% efciency. The heat energy expelled at the low temperature is absorbed by a stream of water that enters the cooling coils at 20 C. If 60 L ows across the coils per second, determine the increase in temperature of the water.

When a gas follows path 123 on the PV diagram in Figure P12.50, 418 J of energy ows into the system by heat and 167 J of work is done on the gas. (a) What is the change in the internal energy of the system? (b) How much energy Q ows into the system if the gas follows path 143? The work done on the gas along this path is 63.0 J. What net work would be done on or by the system if the system followed (c) path 12341? (d) path 14321? (e) What is the change in internal energy of the system in the processes described in parts (c) and (d)?

A substance undergoes the cyclic process shown in Figure P12.51. Work output occurs along path AB while work input is required along path BC, and no work is involved in the constant volume process CA. Energy transfers by heat occur during each process involved in the cycle. (a) What is the work output during process AB? (b) How much work input is required during process BC? (c) What is the net energy input Q during this cycle?

A power plant having a Carnot efciency produces 1 000 MW of electrical power from turbines that take in steam at 500 K and eject water at 300 K into a owing river. The water downstream is 6.00 K warmer due to the output of the plant. Determine the ow rate of the river.

A power plant having a Carnot efciency produces 1 000 MW of electrical power from turbines that take in steam at 500 K and eject water at 300 K into a owing river. The water downstream is 6.00 K warmer due to the output of the plant. Determine the ow rate of the river.

An ideal gas initially at pressure P0, volume V0, and temperature T0 is taken through the cycle described in Figure P12.54. (a) Find the net work done by the gas per cycle in terms of P0 and V0. (b) What is the net energy Q added to the system per cycle? (c) Obtain a numerical value for the net work done per cycle for 1.00 mol of gas initially at 0 C. (Hint: Recall that the work done by the system equals the area under a PV curve.)

One mole of neon gas is heated from 300 K to 420 K at constant pressure. Calculate (a) the energy Q transferred to the gas, (b) the change in the internal energy of the gas, and (c) the work done on the gas. Note that neon has a molar specic heat of c  20.79 J/molK for a constantpressure process.

A 1.0-kg block of aluminum is heated at atmospheric pressure so that its temperature increases from 22 C to 40 C. Find (a) the work done on the aluminum, (b) the energy Q added to the aluminum, and (c) the change in internal energy of the aluminum.

Suppose a heat engine is connected to two energy reservoirs, one a pool of molten aluminum at 660 C and the other a block of solid mercury at 38.9 C. The engine runs by freezing 1.00 g of aluminum and melting 15.0 g of mercury during each cycle. The latent heat of fusion of aluminum is 3.97  105 J/kg, and that of mercury is 1.18  104 J/kg. (a) What is the efciency of this engine? (b) How does the efciency compare with that of a Carnot engine?

One mole of an ideal gas is taken through the cycle shown in Figure P12.58. At point A, the pressure, volume, and temperature are P0, V0, and T0. In terms of R and T0, nd (a) the total energy entering the system by heat per cycle, (b) the total energy leaving the system by heat per cycle, (c) the efciency of an engine operating in this cycle, and (d) the efciency of an engine operating in a Carnot cycle between the temperature extremes for this process. (Hint: Recall that work done on the gas is the negative of the area under a PV curve.)

An electrical power plant has an overall efciency of 15%. The plant is to deliver 150 MW of electrical power to a city, and its turbines use coal as fuel. The burning coal produces steam at 190 C, which drives the turbines. The steam is condensed into water at 25 C by passing through coils that are in contact with river water. (a) How many metric tons of coal does the plant consume each day (1 metric ton  1  103 kg)? (b) What is the total cost of the fuel per year if the delivery price is $8 per metric ton? (c) If the river water is delivered at 20 C, at what minimum rate must it ow over the cooling coils in order that its temperature not exceed 25 C? (Note: The heat of combustion of coal is 7.8  103 cal/g.)

At atmospheric pressure (1.013  105 Pa) and 20.0 C, 1.00 g of water occupies a volume of 1.00 cm3. (a) Find the change in internal energy when the water is heated to the boiling point. (b)When the water is boiled, it becomes 1 671 cm3 of steam. Calculate the change in internal energy for this process. Assume the steam vapor doesnt mix with the surrounding air and that it expands at atmospheric pressure.

A gas is enclosed in a container tted with a piston of cross-sectional area 0.10 m2. The pressure of the gas is maintained at 8 000 Pa while energy is slowly added by heat; as a result, the piston is pushed up a distance of 4.0 cm. (Recall that any process in which the pressure remains constant is an isobaric process.) (a) If 42 J of energy is added to the system by heat during the expansion, what is the change in internal energy of the system? (b) If 42 J of energy is added by heat to the system with the piston clamped in a xed position, what is the work done by the gas? What is the change in its internal energy?

Hydrothermal vents deep on the ocean oor spout water at temperatures as high as 570 C. This temperature is below the boiling point of water because of the immense pressure at that depth. Since the surrounding ocean temperature is at 4.0 C, an organism could use the temperature gradient as a source of energy. (a) Assuming the specic heat of water under these conditions is 1.0 cal/g C, how much energy is released when 1.0 liter of water is cooled from 570 C to 4.0 C? (b) What is the maximum useable energy an organism can extract from this energy source? (Assume that the organism has some internal type of heat engine acting between the two temperature extremes.) (c) Water from these vents contains hydrogen sulde (H2S) at a concentration of 0.90 mmole/liter. Oxidation of one mole of H2S produces 310 kJ of energy. How much energy is available through H2S oxidation of 1.0 L of water?

Suppose you spend 30.0 minutes on a stair-climbing machine, climbing at a rate of 90.0 steps per minute, with each step 8.00 inches high. If you weigh 150 lb and the machine reports that 600 kcal have been burned at the end of the workout, what efciency is the machine using in obtaining this result? If your actual efciency is 0.18, how many kcal did you really burn?

Bend a paper clip back and forth several times and touch it to your lip. Why does it warm up? Repeat the process with a rubber band after it is stretched several times. Notice that it also warms up after stretching. Now take the stretched rubber band from your lip and wait for it to come to equilibrium with the surrounding air. Touch the stretched band to your lip, let it relax to its unstretched length, and note that it becomes cooler. Explain how the second law of thermodynamics helps you understand these results.

You should be able to locate a helium tank, a nitrogen tank, or a CO2 tank in the chemistry or physics department of your university. Ask to use one of these for a short period for the following observation: Learn to open and close the valve or valves correctly. Touch the tanks and nozzle to verify that they are at room temperature. Open a tank for 23 seconds and then close it. Touch the nozzle again, and you will nd that it is very cold. Why? (Hint: The expansion of the gas is nearly adiabatic.)

You can study order and disorder and verify the answer to Quick Quiz 12.5 by rolling a pair of dice 100 times and recording the number of spots appearing on the dice for each throw. Which total comes up most frequently?

Another way to study order and disorder is to toss pennies. Predict in advance which is the more likely occur rence when tossing three pennies: having the pennies ordered with all the same orientation (all heads or all tails) or having them disordered (not all the same). Show that the ordered result includes two out of eight possibilities, while the disordered result includes six out of eight. Toss the three pennies 100 times to see how close you come to this prediction. Repeat the process for four pennies (The all-the-same result should be expected two times in sixteen throws.)