### Problems

## 4.1 Reversible and Irreversible Processes

A tank contains 111.0 g chlorine gas $({\text{Cl}}_{2}),$ which is at temperature $82.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and absolute pressure $5.70\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}.$ The temperature of the air outside the tank is $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. The molar mass of ${\text{Cl}}_{2}$ is 70.9 g/mol. (a) What is the volume of the tank? (b) What is the internal energy of the gas? (c) What is the work done by the gas if the temperature and pressure inside the tank drop to $31.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and $3.80\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$, respectively, due to a leak?

A mole of ideal monatomic gas at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and 1.00 atm is warmed up to expand isobarically to triple its volume. How much heat is transferred during the process?

A mole of an ideal gas at pressure 4.00 atm and temperature 298 K expands isothermally to double its volume. What is the work done by the gas?

After a free expansion to quadruple its volume, a mole of ideal diatomic gas is compressed back to its original volume adiabatically and then cooled down to its original temperature. What is the minimum heat removed from the gas in the final step to restoring its state?

## 4.2 Heat Engines

An engine is found to have an efficiency of 0.40. If it does 200 J of work per cycle, what are the corresponding quantities of heat absorbed and discharged?

In performing 100.0 J of work, an engine discharges 50.0 J of heat. What is the efficiency of the engine?

An engine with an efficiency of 0.30 absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it discharge per cycle?

It is found that an engine discharges 100.0 J while absorbing 125.0 J each cycle of operation. (a) What is the efficiency of the engine? (b) How much work does it perform per cycle?

The temperature of the cold reservoir of the engine is 300 K. It has an efficiency of 0.30 and absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it discharge per cycle?

An engine absorbs three times as much heat as it discharges. The work done by the engine per cycle is 50 J. Calculate (a) the efficiency of the engine, (b) the heat absorbed per cycle, and (c) the heat discharged per cycle.

A coal power plant consumes 100,000 kg of coal per hour and produces 500 MW of power. If the heat of combustion of coal is 30 MJ/kg, what is the efficiency of the power plant?

## 4.3 Refrigerators and Heat Pumps

A refrigerator has a coefficient of performance of 3.0. (a) If it requires 200 J of work per cycle, how much heat per cycle does it remove the cold reservoir? (b) How much heat per cycle is discarded to the hot reservoir?

During one cycle, a refrigerator removes 500 J from a cold reservoir and discharges 800 J to its hot reservoir. (a) What is its coefficient of performance? (b) How much work per cycle does it require to operate?

If a refrigerator discards 80 J of heat per cycle and its coefficient of performance is 6.0, what are (a) the quantity off heat it removes per cycle from a cold reservoir and (b) the amount of work per cycle required for its operation?

A refrigerator has a coefficient of performance of 3.0. (a) If it requires 200 J of work per cycle, how much heat per cycle does it remove the cold reservoir? (b) How much heat per cycle is discarded to the hot reservoir?

## 4.5 The Carnot Cycle

The temperature of the cold and hot reservoirs between which a Carnot refrigerator operates are $\mathrm{-73}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and $270\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, respectively. Which is its coefficient of performance?

Suppose a Carnot refrigerator operates between ${T}_{\text{c}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{T}_{\text{h}}.$ Calculate the amount of work required to extract 1.0 J of heat from the cold reservoir if (a) ${T}_{\text{c}}=7\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, ${T}_{\text{h}}=27\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$; (b) ${T}_{\text{c}}=\mathrm{-73}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, ${T}_{\text{h}}=27\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C;}$ (c) ${T}_{\text{c}}=\mathrm{-173}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, ${T}_{\text{h}}=27\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$; and (d) ${T}_{\text{c}}=\mathrm{-273}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, ${T}_{\text{h}}=27\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$.

A Carnot engine operates between reservoirs at 600 and 300 K. If the engine absorbs 100 J per cycle at the hot reservoir, what is its work output per cycle?

A 500-W motor operates a Carnot refrigerator between $\mathrm{-5}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and $30\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (a) What is the amount of heat per second extracted from the inside of the refrigerator? (b) How much heat is exhausted to the outside air per second?

A Carnot heat pump operates between $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. How much heat is exhausted into the interior of a house for every 1.0 J of work done by the pump?

An engine operating between heat reservoirs at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and $200\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ extracts 1000 J per cycle from the hot reservoir. (a) What is the maximum possible work that engine can do per cycle? (b) For this maximum work, how much heat is exhausted to the cold reservoir per cycle?

Suppose a Carnot engine can be operated between two reservoirs as either a heat engine or a refrigerator. How is the coefficient of performance of the refrigerator related to the efficiency of the heat engine?

A Carnot engine is used to measure the temperature of a heat reservoir. The engine operates between the heat reservoir and a reservoir consisting of water at its triple point. (a) If 400 J per cycle are removed from the heat reservoir while 200 J per cycle are deposited in the triple-point reservoir, what is the temperature of the heat reservoir? (b) If 400 J per cycle are removed from the triple-point reservoir while 200 J per cycle are deposited in the heat reservoir, what is the temperature of the heat reservoir?

What is the minimum work required of a refrigerator if it is to extract 50 J per cycle from the inside of a freezer at $\mathrm{-10}\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and exhaust heat to the air at $25\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$?

## 4.6 Entropy

Two hundred joules of heat are removed from a heat reservoir at a temperature of 200 K. What is the entropy change of the reservoir?

In an isothermal reversible expansion at $27\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, an ideal gas does 20 J of work. What is the entropy change of the gas?

An ideal gas at 300 K is compressed isothermally to one-fifth its original volume. Determine the entropy change per mole of the gas.

What is the entropy change of 10 g of steam at $100\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ when it condenses to water at the same temperature?

A metal rod is used to conduct heat between two reservoirs at temperatures ${T}_{\text{h}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{T}_{\text{c}},$ respectively. When an amount of heat *Q* flows through the rod from the hot to the cold reservoir, what is the net entropy change of the rod, the hot reservoir, the cold reservoir, and the universe?

For the Carnot cycle of Figure 4.12, what is the entropy change of the hot reservoir, the cold reservoir, and the universe?

A 5.0-kg piece of lead at a temperature of $600\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is placed in a lake whose temperature is $15\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. Determine the entropy change of (a) the lead piece, (b) the lake, and (c) the universe.

One mole of an ideal gas doubles its volume in a reversible isothermal expansion. (a) What is the change in entropy of the gas? (b) If 1500 J of heat are added in this process, what is the temperature of the gas?

An ideal monatomic gas is confined to a rigid container. When heat is added reversibly to the gas, its temperature changes from ${T}_{1}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}{T}_{2}.$ (a) How much heat is added? (b) What is the change in entropy of the gas?

(a) A 5.0-kg rock at a temperature of $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is dropped into a shallow lake also at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ from a height of $1.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{m}$. What is the resulting change in entropy of the universe? (b) If the temperature of the rock is $100\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ when it is dropped, what is the change of entropy of the universe? Assume that air friction is negligible (not a good assumption) and that $c=860\phantom{\rule{0.2em}{0ex}}\text{J/kg}\xb7\text{K}$ is the specific heat of the rock.

## 4.7 Entropy on a Microscopic Scale

A copper rod of cross-sectional area $5.0\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ and length 5.0 m conducts heat from a heat reservoir at 373 K to one at 273 K. What is the time rate of change of the universe’s entropy for this process?

Fifty grams of water at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is heated until it becomes vapor at $100\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. Calculate the change in entropy of the water in this process.

Fifty grams of water at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ are changed into vapor at $100\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. What is the change in entropy of the water in this process?

In an isochoric process, heat is added to 10 mol of monoatomic ideal gas whose temperature increases from 273 to 373 K. What is the entropy change of the gas?

Two hundred grams of water at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is brought into contact with a heat reservoir at $80\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. After thermal equilibrium is reached, what is the temperature of the water? Of the reservoir? How much heat has been transferred in the process? What is the entropy change of the water? Of the reservoir? What is the entropy change of the universe?

Suppose that the temperature of the water in the previous problem is raised by first bringing it to thermal equilibrium with a reservoir at a temperature of $40\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and then with a reservoir at $80\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. Calculate the entropy changes of (a) each reservoir, (b) of the water, and (c) of the universe.

Two hundred grams of water at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is brought into contact into thermal equilibrium successively with reservoirs at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, $40\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, $60\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, and $80\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (a) What is the entropy change of the water? (b) Of the reservoir? (c) What is the entropy change of the universe?

(a) Ten grams of ${\text{H}}_{2}\text{O}$ starts as ice at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. The ice absorbs heat from the air (just above $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$) until all of it melts. Calculate the entropy change of the ${\text{H}}_{2}\text{O}$, of the air, and of the universe. (b) Suppose that the air in part (a) is at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ rather than $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ and that the ice absorbs heat until it becomes water at $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. Calculate the entropy change of the ${\text{H}}_{2}\text{O}$, of the air, and of the universe. (c) Is either of these processes reversible?

The Carnot cycle is represented by the temperature-entropy diagram shown below. (a) How much heat is absorbed per cycle at the high-temperature reservoir? (b) How much heat is exhausted per cycle at the low-temperature reservoir? (c) How much work is done per cycle by the engine? (d) What is the efficiency of the engine?

A Carnot engine operating between heat reservoirs at 500 and 300 K absorbs 1500 J per cycle at the high-temperature reservoir. (a) Represent the engine’s cycle on a temperature-entropy diagram. (b) How much work per cycle is done by the engine?

A monoatomic ideal gas (*n* moles) goes through a cyclic process shown below. Find the change in entropy of the gas in each step and the total entropy change over the entire cycle.

A Carnot engine has an efficiency of 0.60. When the temperature of its cold reservoir changes, the efficiency drops to 0.55. If initially ${T}_{\text{c}}=27\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, determine (a) the constant value of ${T}_{\text{h}}$ and (b) the final value of ${T}_{\text{c}}$.

A Carnot engine performs 100 J of work while discharging 200 J of heat each cycle. After the temperature of the hot reservoir only is adjusted, it is found that the engine now does 130 J of work while discarding the same quantity of heat. (a) What are the initial and final efficiencies of the engine? (b) What is the fractional change in the temperature of the hot reservoir?

A Carnot refrigerator exhausts heat to the air, which is at a temperature of $25\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. How much power is used by the refrigerator if it freezes 1.5 g of water per second? Assume the water is at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$.