Solved: Suppose an ideal (Carnot) heat pump could be

A monatomic ideal gas expands from 1.00 m3 to 2.50 m3 at a constant pressure of 2.00 3 105 Pa. Find the change in the internal energy of the gas. (a) 7.50 3 105 J (b) 1.05 3 106 J (c) 4.50 3 105 J (d) 3.00 3 105 J (e) 24.50 3 105 J

A 2.0-mole ideal gas system is maintained at a constant volume of 4.0 liters. If 100 J of thermal energy is transferred to the system, what is the change in the internal energy of the system? (a) 0 (b) 400 J (c) 70 J (d) 100 J (e) 20.100 J

An ideal gas is maintained at a constant pressure of 70.0 kPa during an isobaric process while its volume decreases by 0.20 m3. What is the work done by the system on its environment? (a) 14 kJ (b) 35 kJ (c) 214 kJ (d) 235 kJ (e) 272 kJ

How much net work is done by the gas undergoing the cyclic process illustrated in Figure MCQ12.4? Choose the best estimate. (a) 1 3 105 J (b) 2 3 105 J (c) 3 3 105 J (d) 4 3 105 J (e) 5 3 105 Pa

A diatomic ideal gas expands adiabatically from a volume of 1.00 m3 to a final volume of 3.50 m3. If the initial pressure is 1.00 3 105 Pa, what is the final pressure? (a) 6.62 3 104 Pa (b) 1.24 3 105 Pa (c) 3.54 3 104 Pa (d) 2.33 3 104 Pa (e) 1.73 3 104 Pa

An ideal gas drives a piston as it expands from 1.00 m3 to 2.00 m3 at a constant temperature of 850 K. If there are 390 moles of gas in the piston, how much work does the gas do in displacing the piston? (a) 1.9 3 106 J (b) 2.5 3 106 J (c) 4.7 3 106 J (d) 2.1 3 105 J (e) 3.5 3 106 J

An engine does 15 kJ of work while rejecting 37 kJ to the cold reservoir. What is the efficiency of the engine? (a) 0.15 (b) 0.29 (c) 0.33 (d) 0.45 (e) 1.2

A refrigerator does 18 kJ of work while moving 115 kJ of thermal energy from inside the refrigerator. What is its coefficient of performance? (a) 3.4 (b) 2.8 (c) 8.9 (d) 6.4 (e) 5.2

If an ideal gas is compressed isothermally, which of the following statements is true? (a) Energy is transferred to the gas by heat. (b) No work is done on the gas. (c) The temperature of the gas increases. (d) The internal energy of the gas remains constant. (e) None of those statements is true.

A 1.00-kg block of ice at 0C and 1.0 atm melts completely to water at 0C. Calculate the change of the entropy of the ice during the melting process. (For ice, Lf 5 3.33 3 105 J/kg.) (a) 3 340 J/K (b) 2 170 J/K (c) 23 340 J/K (d) 1 220 J/K (e) 21 220 J/K

A steam turbine operates at a boiler temperature of 450 K and an exhaust temperature of 3.0 3 102 K. What is the maximum theoretical efficiency of this system? (a) 0.24 (b) 0.50 (c) 0.33 (d) 0.67 (e) 0.15

When an ideal gas undergoes an adiabatic expansion, which of the following statements is true? (a) The temperature of the gas doesnt change. (b) No work is done by the gas. (c) No energy is delivered to the gas by heat. (d) The internal energy of the gas doesnt change. (e) The pressure increases.

If an ideal gas undergoes an isobaric process, which of the following statements is true? (a) The temperature of the gas doesnt change. (b) Work is done on or by the gas. (c) No energy is transferred by heat to or from the gas. (d) The volume of the gas remains the same. (e) The pressure of the gas decreases uniformly.

Two different ideal gases, designated A and B, occupy flasks with identical volumes. They have the same initial pressures and temperatures. The same amount of thermal energy Q is transferred to each of them. The final pressure of gas A is greater than the final pressure of gas B. Which of the following statements could explain the discrepancy? (a) Gas A is monatomic; gas B is diatomic. (b) Gas A is diatomic; gas B is monatomic. (c) There are more moles of gas A than gas B. (d) There are more moles of gas B than of gas A. (e) The molar specific heat of gas A is less than that of gas B.

An ideal gas is compressed to half its initial volume by means of several possible processes. Which of the following processes results in the most work done on the gas? (a) isothermal (b) adiabatic (c) isobaric (d) The work done is independent of the process.

A thermodynamic process occurs in which the entropy of a system changes by 26 J/K. According to the second law of thermodynamics, what can you conclude about the entropy change of the environment? (a) It must be 16 J/K or less. (b) It must be equal to 6 J/K. (c) It must be between 6 J/K and 0. (d) It must be 0. (e) It must be 16 J/K or more.

When sufficient thermal energy is added to a substance such as lead, it melts. What happens to the entropy of the substance as it melts? (a) It remains the same. (b) It increases. (c) It decreases. (d) It may increase or decrease, depending on the mass of the substance. (e) It may increase or decrease depending on the process used.

A window air conditioner is placed on a table inside a well-insulated apartment, plugged in and turned on. What happens to the average temperature of the apartment? (a) It increases. (b) It decreases. (c) It remains constant. (d) It increases until the unit warms up and then decreases. (e) The answer depends on the initial temperature of the apartment.

An ideal gas is compressed from a volume of Vi 5 5.00 L to a volume of Vf 5 3.00 L while in thermal contact with a heat reservoir at T 5 295 K as in Figure P12.19. During the compression process, the piston moves down a distance of d 5 0.130 m under the action of an average external force of F 5 25.0 kN. Find (a) the work done on the gas, (b) the change in internal energy of the gas, and (c) the thermal energy exchanged between the gas and the reservoir. (d) If the gas is thermally insulated so no thermal energy could be exchanged, what would happen to the temperature of the gas during the compression?

A system consisting of 0.025 6 moles of a diatomic ideal gas is taken from state A to state C along the path in Figure P12.20. (a) How much work is done on the gas during this process? (b) What is the lowest temperature of the gas during this process, and where does it occur? (c) Find the change in internal energy of the gas and (d) the energy delivered to the gas in going from A to C. Hint: For part (c), adapt the equation in the remarks of Example 12.9 to a diatomic ideal gas.

An ideal monatomic gas expands isothermally from 0.500 m3 to 1.25 m3 at a constant temperature of 675 K. If the initial pressure is 1.00 3 105 Pa, find (a) the work done on the gas, (b) the thermal energy transfer Q, and (c) the change in the internal energy.

An ideal gas expands at constant pressure. (a) Show that PDV 5 nRDT. (b) If the gas is monatomic, start from the definition of internal energy and show that DU 5 32Wenv, where Wenv is the work done by the gas on its environment. (c) For the same monatomic ideal gas, show with the first law that Q 5 52 Wenv. (d) Is it possible for an ideal gas to expand at constant pressure while exhausting thermal energy? Explain.

An ideal monatomic gas is contained in a vessel of constant volume 0.200 m3. The initial temperature and pressure of the gas are 300 K and 5.00 atm, respectively. The goal of this problem is to find the temperature and pressure of the gas after 16.0 kJ of thermal energy is supplied to the gas. (a) Use the ideal gas law and initial conditions to calculate the number of moles of gas in the vessel. (b) Find the specific heat of the gas. (c) What is the work done by the gas during this process? (d) Use the first law of thermodynamics to find the change in internal energy of the gas. (e) Find the change in temperature of the gas. (f) Calculate the final temperature of the gas. (g) Use the ideal gas expression to find the final pressure of the gas.

Consider the cyclic process described by Figure P12.24. If Q is negative for the process BC and DU is negative for the process CA, determine the signs of Q, W, and DU associated with each process.

A 5.0-kg block of aluminum is heated from 20C to 90C at atmospheric pressure. Find (a) the work done by the aluminum, (b) the amount of energy transferred to it by heat, and (c) the increase in its internal energy.

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 final state, the gas is at a pressure of 1.50 atm and a volume of 0.800 L, and its internal energy equals 182 J. For the paths IAF, IBF, and IF in Figure P12.26, calculate (a) the work done on the gas and (b) the net energy transferred to the gas by heat in the process.

Consider the Universe to be an adiabatic expansion of atomic hydrogen gas. (a) Use the ideal gas law and Equation 12.8a to show that TV g21 5 C, where C is a constant. (b) The current Universe extends at least 15 billion light-years in all directions (1.4 3 1026 m), and the current temperature of the Universe is 2.7 K. Estimate the temperature of the Universe when it was the size of a nutshell, with a radius of 2 cm. (For this calculation, assume the Universe is spherical.)

Suppose the Universe is considered to be an ideal gas of hydrogen atoms expanding adiabatically. (a) If the density of the gas in the Universe is one hydrogen atom per cubic meter, calculate the number of moles per unit volume (n/V). (b) Calculate the pressure of the Universe, taking the temperature of the Universe as 2.7 K. (c) If the current radius of the Universe is 15 billion light-years (1.4 3 1026 m), find the pressure of the Universe when it was the size of a nutshell, with radius 2.0 3 1022 m. Be careful: Calculator overflow can occur.

A gas increases in pressure from 2.00 atm to 6.00 atm at a constant volume of 1.00 m3 and then expands at constant pressure to a volume of 3.00 m3 before returning to its initial state as shown in Figure P12.29. How much work is done in one cycle?

An ideal gas expands at a constant pressure of 6.00 3 105 Pa from a volume of 1.00 m3 to a volume of 4.00 m3 and then is compressed to one-third that pressure and a volume of 2.50 m3 as shown in Figure P12.30 before returning to its initial state. How much work is done in taking a gas through one cycle of the process shown in the figure?

A heat engine operates between a reservoir at 25C and one at 375C. What is the maximum efficiency possible for this engine?

A heat engine is being designed to have a Carnot efficiency of 65% when operating between two heat reservoirs. (a) If the temperature of the cold reservoir is 20C, what must be the temperature of the hot reservoir? (b) Can the actual efficiency of the engine be equal to 65%? Explain.

The work done by an engine equals one-fourth the energy it absorbs from a reservoir. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir?

In each cycle of its operation, a heat engine expels 2 400 J of energy and performs 1 800 J of mechanical work. (a) How much thermal energy must be added to the engine in each cycle? (b) Find the thermal efficiency of the engine.

One of the most efficient engines ever built is a coalfired steam turbine engine in the Ohio River valley, driving an electric generator as it operates between 1 870C and 430C. (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 3 105 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 fires one 2.40-g bullet at 320 m/s with an energy efficiency of 1.10%. Assume the body of the gun absorbs all 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 1.70 kJ from a hot reservoir at 277C and expels 1.20 kJ to a cold reservoir at 27C in each cycle. (a) What is the engines efficiency? (b) How much work is done by the engine in each cycle? (c) What is the power output of the engine if each cycle lasts 0.300 s?

A heat pump has a coefficient of performance of 3.80 and operates with a power consumption of 7.03 3 103 W. (a) How much energy does the heat pump deliver into a home during 8.00 h of continuous operation? (b) How much energy does it extract from the outside air in 8.00 h?

A freezer has a coefficient of performance of 6.30. The freezer is advertised as using 457 kW-h/y. (a) On average, how much energy does the freezer use in a single day? (b) On average, how much thermal energy is removed from the freezer each day? (c) What maximum mass of water at 20.0C could the freezer freeze in a single day? Note: One kilowatt-hour (kW-h) is an amount of energy equal to operating a 1-kW appliance for one hour.

Suppose an ideal (Carnot) heat pump could be constructed. (a) Using Equation 12.15, obtain an expression for the coefficient of performance for such a heat pump in terms of Th and Tc. (b) Would such a heat pump work better if the difference in the operating temperatures were greater or were smaller? (c) Compute the coefficient of performance for such a heat pump if the cold reservoir is 50.0C and indoor temperature is 70.0C.

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

A power plant has been proposed that would make use of the temperature gradient in the ocean. The system is to operate between 20.0C (surface water temperature) and 5.00C (water temperature at a depth of about 1 km). (a) What is the maximum efficiency 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 part (a), do you think such a system is worthwhile (considering that there is no charge for fuel)?

A certain nuclear power plant has an electrical power output of 435 MW. The rate at which energy must be supplied to the plant is 1 420 MW. (a) What is the thermal efficiency of the power plant? (b) At what rate is thermal energy expelled by the plant?

A heat engine operates in a Carnot cycle between 80.0C and 350C. 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 Styrofoam cup holding 125 g of hot water at 1.00 3 102C cools to room temperature, 20.0C. What is the change in entropy of the room? (Neglect the specific heat of the cup and any change in temperature of the room.)

A 65-g ice cube is initially at 0.0C. (a) Find the change in entropy of the cube after it melts completely at 0.0C. (b) What is the change in entropy of the environment in this process? Hint: The latent heat of fusion for water is 3.33 3 105 J/kg.

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 5 0C. 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 100C as it changes to steam at 100C?

A 70.0-kg log falls from a height of 25.0 m into a lake. If the log, the lake, and the air are all at 300 K, find the change in entropy of the Universe during this process.

If you roll a pair of dice, what is the total number of ways in which you can obtain (a) a 12? (b) a 7?

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?

When an aluminum bar is temporarily connected between a hot reservoir at 725 K and a cold reservoir at 310 K, 2.50 kJ of energy is transferred by heat from the hot reservoir to the cold reservoir. In this irreversible process, calculate the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe, neglecting any change in entropy of the aluminum rod. (d) Mathematically, why did the result for the Universe in part (c) have to be positive?

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?

This is a symbolic version of Problem 52. When a metal bar is temporarily connected between a hot reservoir at Th and a cold reservoir at Tc, the energy transferred by heat from the hot reservoir to the cold reservoir is Qh. In this irreversible process, find expressions for the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe.

On a typical day, a 65-kg man sleeps for 8.0 h, does light chores for 3.0 h, walks slowly for 1.0 h, and jogs at moderate pace for 0.5 h. What is the change in his internal energy for all these activities?

A weightlifter has a basal metabolic rate of 80.0 W. As he is working out, his metabolic rate increases by about 650 W. (a) How many hours does it take him to work off a 450-Calorie bagel if he stays in bed all day? (b) How long does it take him if hes working out? (c) Calculate the amount of mechanical work necessary to lift a 120-kg barbell 2.00 m. (d) He drops the barbell to the floor and lifts it repeatedly. How many times per minute must he repeat this process to do an amount of mechanical work equivalent to his metabolic rate increase of 650 W during exercise? (e) Could he actually do repetitions at the rate found in part (d) at the given metabolic level? Explain.

Sweating is one of the main mechanisms with which the body dissipates heat. Sweat evaporates with a latent heat of 2 430 kJ/kg at body temperature, and the body can produce as much as 1.5 kg of sweat per hour. If sweating were the only heat dissipation mechanism, what would be the maximum sustainable metabolic rate, in watts, if 80% of the energy used by the body goes into waste heat?

A Carnot engine operates between the temperatures Th 5 100C and Tc 5 20C. By what factor does the theoretical efficiency increase if the temperature of the hot reservoir is increased to 550C?

A 1 500-kW heat engine operates at 25% efficiency. The heat energy expelled at the low temperature is absorbed by a stream of water that enters the cooling coils at 20C. If 60 L flows across the coils per second, determine the increase in temperature of the water.

A Carnot engine operates between 100C and 20C. How much ice can the engine melt from its exhaust after it has done 5.0 3 104 J of work?

A substance undergoes the cyclic process shown in Figure P12.61. 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?

When a gas follows path 123 on the PV diagram in Figure P12.62, 418 J of energy flows into the system by heat and 2167 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 flows into the system if the gas follows path 143? The work done on the gas along this path is 263.0 J. What net work would be done on or by the system if the system followed (c) path 12341 and (d) path 14321? (e) What is the change in internal energy of the system in the processes described in parts (c) and (d)?

A 100-kg steel support rod in a building has a length of 2.0 m at a temperature of 20C. The rod supports a hanging load of 6 000 kg. Find (a) the work done on the rod as the temperature increases to 40C, (b) the energy Q added to the rod (assume the specific heat of steel is the same as that for iron), and (c) the change in internal energy of the rod.

An ideal gas initially at pressure P0, volume V0, and temperature T0 is taken through the cycle described in Figure P12.64. (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 0C. 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 specific heat of c 5 20.79 J/mol ? K for a constant-pressure process.

Every second at Niagara Falls, approximately 5 000 m3 of water falls a distance of 50.0 m. What is the increase in entropy per second due to the falling water? Assume the mass of the surroundings is so great that its temperature and that of the water stay nearly constant at 20.0C. Also assume a negligible amount of water evaporates.

A cylinder containing 10.0 moles of a monatomic ideal gas expands from _ to _ along the path shown in Figure P12.67. (a) Find the temperature of the gas at point A and the temperature at point _. (b) How much work is done by the gas during this expansion? (c) What is the change in internal energy of the gas? (d) Find the energy transferred to the gas by heat in this process.

Two moles of molecular hydrogen (H2) react with 1 mole of molecular oxygen (O2) to produce 2 moles of water (H2O) together with an energy release of 241.8 kJ/mole of water. Suppose a spherical vessel of radius 0.500 m contains 14.4 moles of H2 and 7.2 moles of O2 at 20.0C. (a) What is the initial pressure in the vessel? (b) What is the initial internal energy of the gas? (c) Suppose a spark ignites the mixture and the gases burn completely into water vapor. How much energy is produced? (d) Find the temperature and pressure of the steam, assuming its an ideal gas. (e) Find the mass of steam and then calculate the steams density. (f) If a small hole were put in the sphere, what would be the initial exhaust velocity of the exhausted steam if spewed out into a vacuum? (Use Bernoullis equation.)

Suppose you spend 30.0 minutes on a stairclimbing 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 efficiency is the machine using in obtaining this result? If your actual efficiency is 0.18, how many kcal did you actually burn?

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

An electrical power plant has an overall efficiency 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 190C, which drives the turbines. The steam is condensed into water at 25C 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 5 1 3 103 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 20C, at what minimum rate must it flow over the cooling coils so that its temperature doesnt exceed 25C? Note: The heat of combustion of coal is 7.8 3 103 cal/g.

A diatomic ideal gas expands from a volume of VA 5 1.00 m3 to VB 5 3.00 m3 along the path shown in Figure P12.72. If the initial pressure is PA 5 2.00 3 105 Pa and there are 87.5 mol of gas, calculate (a) the work done on the gas during this process, (b) the change in temperature of the gas, and (c) the change in internal energy of the gas. (d) How much thermal energy is transferred to the system?