Heat and Thermodynamics

M. W. Zemansky, Richard H. Dittman

Chapter 6

The Second Law of Thermodynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that the thermal efficiency of an ideal Otto cycle is given by
$$
\eta=1-\frac{1}{\eta^{\gamma-1}}
$$
where the ratio $r=V_{1} / V_{2}$ is called either the compression ratio or the expansion ratio for a gasoline engine. In practice, $r$ cannot be made greater than about 10 , because if $r$ is larger, then the rise in temperature upon compression of the mixture of gasoline and air is great enough to cause combustion before the discharge of the spark. This is called preignition. Take $r$ equal to 9 and $\gamma$ equal to approximately $1.3$ (because it is a mixture) and calculate the thermal efficiency.

Supratim Pal

Supratim Pal

Numerade Educator

Problem 2

Show that the thermal efficiency of an ideal Diesel cycle is given by
$$
\eta=1-\frac{1}{\gamma} \frac{\left(1 / r_{E}\right)^{7}-\left(1 / r_{C}\right)^{7}}{\left(1 / r_{E}\right)-\left(1 / r_{C}\right)}
$$
where the ratio $r_{C}=V_{1} / V_{2}$ is called the compression ratio and the ratio $r_{E}=V_{3} / V_{2}$ is called the expansion ratio for a diesel engine. The compression ratio of a diesel engine is much larger than that of a gasoline engine, because there is no preignition as only air is being compressed. Take $r_{C}=20, r_{E}=5$, and $\gamma=1.4$ and calculate the thermal efficiency.

Nick Johnson

Numerade Educator

Problem 3

Figure $\mathrm{P} 6-1$ represents a simplified $P V$ diagram of the Joule ideal-gas cycle. All processes are quasi-static, and $C_{P}$ is constant. Prove that the thermal efficiency of an engine performing this cycle is
$$
\eta=1-\left(\frac{P_{1}}{P_{2}}\right)^{(\gamma-1) / \gamma}
$$

Penny Riley

Numerade Educator

Problem 4

Figure $\mathrm{P} 6-2$ represents a simplified $P V$ diagram of the Sargent ideal-gas cycle. All processes are quasi-static, and the heat capacities are constant. Prove that the thermal efficiency of an engine performing this cycle is
$$
\eta=1-\gamma \frac{T_{4}-T_{1}}{T_{3}-T_{2}}
$$

Penny Riley

Numerade Educator

Problem 5

Figure P6-3 represents an imaginary ideal-gas cycle. Assuming constant heat capacities, show that the thermal efficiency is
$$
\eta=1-\gamma \frac{\left(V_{1} / V_{2}\right)-1}{\left(P_{3} / P_{2}\right)-1}
$$

Penny Riley

Numerade Educator

Problem 6

An imaginary ideal-gas engine operates in a cycle, which forms a rectangle with sides parallel to the axes of a $P V$ diagram. Call $P_{1}$ and $P_{2}$ the lower and higher pressures, respectively; call $V_{1}$ and $V_{2}$ the lower and higher volumes, respectively.
(a) Calculate the work done in one cycle.
(b) Indicate which parts of the cycle involve heat flow into the gas, and calculate the amount of heat flowing into the gas in one cycle. (Assume constant heat capacities.)
(c) Show that the efficiency of this engine is
$$
\eta=\frac{\gamma-1}{\frac{\gamma P_{2}}{P_{2}-P_{1}}+\frac{V_{1}}{V_{2}-V_{1}}}
$$

Surendra Kumar

Numerade Educator

Problem 7

A vessel contains $10^{-3} \mathrm{~m}^{3}$ of helium gas at $3 \mathrm{~K}$ and $10^{3} \mathrm{~Pa}$. Take the zero of internal energy of helium to be at this state.
(a) The temperature is raised at constant volume to $300 \mathrm{~K}$. Assuming helium to behave like an ideal monatomic gas, how much heat is absorbed, and what is the internal energy of the helium? Can this energy be regarded as the result of heating or working?
(b) The helium is now expanded adiabatically to $3 \mathrm{~K}$. How much work is done, and what is the new internal energy? Has heat been converted to work without compensation, thus violating the second law?
(c) The helium is now compressed isothermally to its original volume. What are the quantities of heat and work in this process? What is the thermal efficiency of the cycle? Plot the cycle on a $P V$ diagram.

Mihajlo Grcic

Numerade Educator

Problem 8

In the tropics, the water near the surface is warmer than the deep water. Would an engine operating between these two levels violate the second law? Why?

Zachary Warner

Numerade Educator

Problem 9

Would a nuclear power plant violate either the first law or the second law of thermodynamics? Explain.

Shoukat Ali

Other Schools

Problem 10

A storage battery is connected to a motor, which is used to lift a weight. The battery remains at constant temperature by receiving heat from the outside air. Is this a violation of the second law? Why?

Vipender Yadav

Numerade Educator

Problem 11

A convenient measure of the performance of a refrigerator is expressed by the coefficient of performance $\omega$, which is the ratio of the heat extracted from the low-temperature resevoir to the work done on the refrigerant. Unlike the thermal efficiency $\eta$, $\omega$ may be considerably larger than unity. Derive an expression for the heat rejected to the high-temperature reservoir. Such a refrigerator is called a "heat pump" and can warm a house in winter by refrigerating the ground, outside air, or water supplied in the mains. Assume a value of 5 for the coefficient of performance and comment on the effectiveness of a heat pump.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

There are many paramagnetic solids that have internal energies which depend only on temperature, like an ideal gas. In an isothermal decrease of the magnetic field, heat is absorbed from one reservoir and converted completely into work. Is this a violation of the second law? Explain.

Vipender Yadav

Numerade Educator

Problem 13

Prove that it is impossible for two reversible adiabatics to intersect. (Hint: Assume that they do intersect and complete the cycle with an isothermal. Show that the performance of this cycle violates the second law.)

Ashok Prajapati

Ashok Prajapati

Numerade Educator