Chapter 4: The Second Law of ThermodynamicsContent check — University Physics 2, Chapter 4: The Second Law of Thermodynamics

What is the efficiency of a perfect heat engine? What is the coefficient of performance of a perfect refrigerator?

Show that $Q_{\text{h}}-$ $Q_{\text{h}}^{\prime }=$ $Q_{\text{c}}-$ $Q_{\text{c}}^{\prime }$ for the hypothetical engine of Figure 4.10(b).

A Carnot engine operates between reservoirs at $400\text{°C}$ and $30\text{°C}$. (a) What is the efficiency of the engine? (b) If the engine does 5.0 J of work per cycle, how much heat per cycle does it absorb from the high-temperature reservoir? (c) How much heat per cycle does it exhaust to the cold-temperature reservoir? (d) What temperatures at the cold reservoir would give the minimum and maximum efficiency?

A Carnot refrigerator operates between two heat reservoirs whose temperatures are $0\text{°}\text{C}$ and $25\text{°}\text{C}$. (a) What is the coefficient of performance of the refrigerator? (b) If 200 J of work are done on the working substance per cycle, how much heat per cycle is extracted from the cold reservoir? (c) How much heat per cycle is discarded to the hot reservoir?

In Example 4.7, 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?

State an example of a process that occurs in nature that is as close to reversible as it can be.

If the refrigerator door is left open, what happens to the temperature of the kitchen?

In the text, we showed that if the Clausius statement is false, the Kelvin statement must also be false. Now show the reverse, such that if the Kelvin statement is false, it follows that the Clausius statement is false.

Discuss the practical advantages and disadvantages of heat pumps and electric heating.

Speculate as to why nuclear power plants are less efficient than fossil-fuel plants based on temperature arguments.

To increase the efficiency of a Carnot engine, should the temperature of the hot reservoir be raised or lowered? What about the cold reservoir?

What type of processes occur in a Carnot cycle?

Is it possible for a system to have an entropy change if it neither absorbs nor emits heat during a reversible transition? What happens if the process is irreversible?

Discuss the entropy changes in the systems of Exercise 4.16 in terms of disorder.

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

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?

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

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?

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 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?

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?

The temperature of the cold and hot reservoirs between which a Carnot heat pump operates are $\mathrm{-73}\text{°}\text{C}$ and $270\text{°C}$, respectively. Which is its coefficient of performance?

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?

Sketch a Carnot cycle on a temperature-volume diagram.

An engine operating between heat reservoirs at $20\text{°}\text{C}$ and $200\text{°}\text{C}$ 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?

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?

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?

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

A metal rod is used to conduct heat between two reservoirs at temperatures $T_{\text{h}}\text{and}{T}_{\text{c}},$ 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?

A 5.0-kg piece of lead at a temperature of $600\text{°C}$ is placed in a lake whose temperature is $15\text{°C}$. Determine the entropy change of (a) the lead piece, (b) the lake, and (c) the universe.

An ideal monatomic gas is confined to a rigid container. When heat is added reversibly to the gas, its temperature changes from $T_1\text{to}{T}_2.$ (a) How much heat is added? (b) What is the change in entropy of the gas?

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?

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 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 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 $0\text{°}\text{C}$, 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 $5.0\times {10}^{\mathrm{-3}}{\text{m}}^3$ and 0.15 m³, respectively. The engine operates between two thermal baths of temperature 500 K and 300 K. (a) Find the values of volume at B and D. (b) How much heat is absorbed by the gas in the AB isothermal expansion? (c) How much work is done by the gas in the AB isothermal expansion? (d) How much heat is given up by the gas in the CD isothermal compression? (e) How much work is done by the gas in the CD isothermal compression? (f) How much work is done by the gas in the BC adiabatic expansion? (g) How much work is done by the gas in the DA adiabatic compression? (h) Find the value of efficiency of the engine based on the net work and heat input. Compare this value to the efficiency of a Carnot engine based on the temperatures of the two baths.

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 $550\text{°}\text{C}$ and $20\text{°}\text{C}$ 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 $250\text{°}\text{C}$ is dropped into 1.0 kg of water at $20\text{°}\text{C}$. 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 $dU=TdS-dW$. (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

$\text{Δ}S=$ $n{C}_v\text{ln}\frac{T_2}{T_1}+$ $nR\text{ln}\frac{V_2}{V_1}$.

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

$\text{Δ}S=$ $n{C}_p\text{ln}\frac{T_2}{T_1}-$ $nR\text{ln}\frac{p_2}{p_1}$.

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.

An ideal diesel cycle is shown below. This cycle consists of five strokes. In this case, only air is drawn into the chamber during the intake stroke OA. The air is then compressed adiabatically from state A to state B, raising its temperature high enough so that when fuel is added during the power stroke BC, it ignites. After ignition ends at C, there is a further adiabatic power stroke CD. Finally, there is an exhaust at constant volume as the pressure drops from $p_D$ to $p_A$, followed by a further exhaust when the piston compresses the chamber volume to zero.

(a) Use $W=Q_1-Q_2$, $Q_1=$ $n{C}_p(T_C-T_B)$, and $Q_2=$ $n{C}_v(T_D-T_A)$ to show that $e=$ $\frac{W}{Q_1}=$ $1-$ $\frac{T_D-T_A}{\gamma (T_C-T_B)}$.

(b) Use the fact that $A\to B$ and $C\to D$ are adiabatic to show that

$e=$ $1-$ $\frac{1}{\gamma }\frac{{\left(\frac{V_C}{V_D}\right)}^{\gamma }-{\left(\frac{V_B}{V_A}\right)}^{\gamma }}{\left(\frac{V_C}{V_D}\right)-\left(\frac{V_B}{V_A}\right)}$.

(c) Since there is no preignition (remember, the chamber does not contain any fuel during the compression), the compression ratio can be larger than that for a gasoline engine. Typically, $V_A\text{/}{V}_B=$ $15\text{and}{V}_D\text{/}{V}_C=$ $5$. For these values and $\gamma =1.4,$ show that $\epsilon =0.56$, or an efficiency of $56\text{\%}$. Diesel engines actually operate at an efficiency of about $30\text{\%}\text{to}35\text{\%}$ compared with $25\text{\%}\text{to}30\text{\%}$ for gasoline engines.

Derive a formula for the coefficient of performance of a refrigerator using an ideal gas as a working substance operating in the cycle shown below in terms of the properties of the three states labeled 1, 2, and 3.

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