Problems

A tank contains 111.0 g chlorine gas $({\text{Cl}}_2),$ which is at temperature $82.0\text{°C}$ and absolute pressure $5.70\times {10}^5\text{Pa}.$ The temperature of the air outside the tank is $20.0\text{°C}$. 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 $31.0\text{°C}$ and $3.80\times {10}^5\text{Pa}$, respectively, due to a leak?

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?

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?

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?

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?

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?

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?

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?

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

A 500-W motor operates a Carnot refrigerator between $\mathrm{-5}\text{°}\text{C}$ and $30\text{°C}$. (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 $0\text{°C}$ and $20\text{°C}$. How much heat is exhausted into the interior of a house for every 1.0 J of work done by the pump?

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?

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 $\mathrm{-10}\text{°C}$ and exhaust heat to the air at $25\text{°C}$?

In an isothermal reversible expansion at $27\text{°C}$, an ideal gas does 20 J of work. What is the entropy change of the gas?

What is the entropy change of 10 g of steam at $100\text{°C}$ when it condenses to water at the same temperature?

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

(a) A 5.0-kg rock at a temperature of $20\text{°C}$ is dropped into a shallow lake also at $20\text{°C}$ from a height of $1.0\times {10}^3\text{m}$. What is the resulting change in entropy of the universe? (b) If the temperature of the rock is $100\text{°C}$ 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 $c=860\text{J/kg}·\text{K}$ is the specific heat of the rock.

Fifty grams of water at $20\text{°C}$ is heated until it becomes vapor at $100\text{°C}$. Calculate 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?

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 $40\text{°C}$ and then with a reservoir at $80\text{°C}$. Calculate the entropy changes of (a) each reservoir, (b) of the water, and (c) of the universe.

(a) Ten grams of ${\text{H}}_2\text{O}$ starts as ice at $0\text{°C}$. The ice absorbs heat from the air (just above $0\text{°C}$) 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 $20\text{°C}$ rather than $0\text{°C}$ and that the ice absorbs heat until it becomes water at $20\text{°C}$. 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?

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 Carnot engine has an efficiency of 0.60. When the temperature of its cold reservoir changes, the efficiency drops to 0.55. If initially $T_{\text{c}}=27\text{°C}$, determine (a) the constant value of $T_{\text{h}}$ and (b) the final value of $T_{\text{c}}$.

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