4.7 Entropy on a Microscopic Scale

In the example, the spontaneous flow of heat from a hot object to a cold object results in a net increase in entropy of the universe. Discuss how this result can be related to an increase in disorder of the system.

A quantity of heat Q is absorbed from a reservoir at a temperature $T_{\text{h}}$ by a cooler reservoir at a temperature $T_{\text{c}} .$ What is the entropy change of the hot reservoir, the cold reservoir, and the universe?

A 50-g copper piece at a temperature of $20 °\text{C}$ is placed into a large insulated vat of water at $100 °\text{C}$. (a) What is the entropy change of the copper piece when it reaches thermal equilibrium with the water? (b) What is the entropy change of the water? (c) What is the entropy change of the universe?

Discuss the entropy changes in the systems of the exercise in terms of disorder.

A copper rod of cross-sectional area $5.0 \text{cm}^{2}$ 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 $0 °\text{C}$ are changed into vapor at $100 °\text{C}$. What is the change in entropy of the water in this process?

Two hundred grams of water at $0 °\text{C}$ is brought into contact with a heat reservoir at $80 °\text{C}$. 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?

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

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 300-W heat pump operates between the ground, whose temperature is $0 ° \text{C}$, and the interior of a house at $22 °\text{C}$. What is the maximum amount of heat per hour that the heat pump can supply to the house?

A Carnot engine employs 1.5 mol of nitrogen gas as a working substance, which is considered as an ideal diatomic gas with $\gamma = 7 / 5$ at the working temperatures of the engine. The Carnot cycle goes in the cycle ABCDA with AB being an isothermal expansion. The volume at points A and C of the cycle are $5.0 \times 10^{−3} \text{m}^{3}$ and 0.15 m³, respectively. The engine operates between two thermal baths of temperature 500 K and 300 K.…

A system consisting of 20.0 mol of a monoatomic ideal gas is cooled at constant pressure from a volume of 50.0 L to 10.0 L. The initial temperature was 300 K. What is the change in entropy of the gas?

A Carnot engine operates between $550 ° \text{C}$ and $20 ° \text{C}$ baths and produces 300 kJ of energy in each cycle. Find the change in entropy of the (a) hot bath and (b) cold bath, in each Carnot cycle?

A 0.50-kg piece of aluminum at $250 ° \text{C}$ is dropped into 1.0 kg of water at $20 ° \text{C}$. After equilibrium is reached, what is the net entropy change of the system?

A heat engine operates between two temperatures such that the working substance of the engine absorbs 5000 J of heat from the high-temperature bath and discharges 3000 J to the low-temperature bath. The rest of the energy is converted into mechanical energy of the turbine. Find (a) the amount of work produced by the engine and (b) the efficiency of the engine.

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?

A Carnot engine working between two heat baths of temperatures 600 K and 273 K completes each cycle in 5 sec. In each cycle, the engine absorbs 10 kJ of heat. Find the power of the engine.

(a) An infinitesimal amount of heat is added reversibly to a system. By combining the first and second laws, show that $d U = T d S - d W$. (b) When heat is added to an ideal gas, its temperature and volume change from $T_{1} \text{and} V_{1} \text{to} T_{2} \text{and} V_{2}$. Show that the entropy change of n moles of the gas is given by

$\Delta S = n C_{v} \text{ln} \frac{T_{2}}{T_{1}} + n R \text{ln} \frac{V_{2}}{V_{1}}$.

With the help of the two preceding problems, show that $\Delta S$ between states 1 and 2 of n moles an ideal gas is given by

$\Delta S = n C_{p} \text{ln} \frac{T_{2}}{T_{1}} - n R \text{ln} \frac{p_{2}}{p_{1}}$.

A diatomic ideal gas is brought from an initial equilibrium state at $p_{1} = 0.50 \text{atm}$ and $T_{1} = 300 \text{K}$ to a final stage with $p_{2} = 0.20 \text{atm}$ and $T_{1} = 500 \text{K} .$ Use the results of the previous problem to determine the entropy change per mole of the gas.

A Carnot refrigerator, working between $0 ° \text{C}$ and $30 ° \text{C}$ is used to cool a bucket of water containing $10^{−2} \text{m}^{3}$ of water at $30 ° \text{C}$ to $5 ° \text{C}$ in 2 hours. Find the total amount of work needed.