Chapter Questions
Take an ideal monatomic gas $(\gamma=5)$ around the Carnot cycle, where $T_{H}=600 \mathrm{~K}$ and $T_{L}=300 \mathrm{~K}$. Point 1 at the beginning of the adiabatic compression has pressure $P_{1}=P_{0}$ (atmospheric pressure) and volume $V_{1}=50$ liters. Point 3 has a volume $V_{3}=75$ liters. The resulting Carnot cycle is shown in Fig. P7.1. Calculate the values of volume and pressure at all four points, which have the same meaning as those in Fig. $7-8$.
Take an ideal monatomic gas $\left(\gamma=\frac{5}{3}\right)$ around the Carnot cycle, where point 1 at the beginning of the adiabatic compression has pressure $P_{1}=P_{0}$ (atmospheric pressure), volume $V_{1}=13$ liters, and temperature $T_{1}=300 \mathrm{~K}$. Point 3 has pressure $P_{3}=2 P_{0}$ and volume $V_{3}=26$ liters. The resulting Carnot cycle is shown in Fig. $\mathrm{P} 7-2$. Calculate the values of volume and pressure at all four points, which have the same meaning as those in Fig. $7-8$.
An inventor claims to have developed an engine that takes in $100,000 \mathrm{Btu}$ at a temperature of $400 \mathrm{~K}$, rejects $40,000 \mathrm{Btu}$ at a temperature of $200 \mathrm{~K}$, and delivers $15 \mathrm{~kW} \cdot \mathrm{h}$ of work. Would you advise investing money to put this engine on the market?
A Carnot engine absorbs $100 \mathrm{~J}$ of heat from a reservoir at the temperature of the normal boiling point of water and rejects heat to a reservoir at the temperature of the triple point of water. Find the heat rejected, the work done by the engine, and the thermal efficiency.
Which is the more effective way to increase the thermal efficiency of a Carnot engine:to increase $T_{H}$, keeping $T_{L}$ constant; or to decrease $T_{L}$, keeping $T_{H}$ constant?
Imagine an irreversible engine $I$ and a Carnot engine $R$ operating between the same two reservoirs. Suppose that they absorb different amounts of heat from the hightemperature reservoir, perform different amounts of work, but reject the same amounts of heat to the low-temperature reservoir. Prove Carnot's theorem with the aid of the Kelvin-Planck statement of the second law.
In Sec. $7.4$, suppose that engine $I$ executes an irreversible cycle, and assume that $\eta_{I}=\eta_{R} .$ Show that this assumption leads to a result that is inconsistent with the irreversibility of $I$, and, therefore, that $\eta_{I}<\eta_{R}$.
Draw a symbolic diagram of a set of Carnot engines with the following characteristics: Each engine absorbs the heat rejected by the preceding onc at the temperature at which it was rejected, and each engine delivers the same amount of work. Show that the temperature intervals between which these engines operate are all equal.
Take a gas whose equation of state is $P(v-b)=R \theta$ and whose heat capacity $C_{V}$ is a function of $\theta$ only through a Carnot cycle, and prove that $\theta=T$.
The initial state of $0.1$ mol of an ideal monatomic gas is $P_{0}=32 \mathrm{~Pa}$ and $V_{0}=8 \mathrm{~m}^{3}$. The final state is $P_{1}=1 \mathrm{~Pa}$ and $V_{1}=64 \mathrm{~m}^{3}$. Suppose that the gas undergoes a process along a straight line joining these two states with an equation $P=a V+b$, where $a=-31 / 56$ and $b=255 / 7$. Plot this straight line to scale on a $P V$ diagram. Calculate:(a) Temperature $T$ as a function of $V$ along the straight line.(b) The value of $V$ at which $T$ is a maximum.(c) The values of $T_{0}, T_{\max }$, and $T_{1}$.(d) The heat $Q$ transferred from the volume $V_{0}$ to any other volume $V$ along the straight line.(e) The values of $P$ and $V$ at which $Q$ is a maximum.(f) The heat transferred along the line from $V_{0}$ to $V$ when $Q$ is a maximum.(g) The heat transferred from $V$ at maximum $Q$ to $V_{1}$.
Show that the two states specified in Prob. $7.10$ lie on an adiabatic curve. A cycle described by J. Willis and D. Kirwan, and called the "Sadly Cannot" cycle, is obtained by proceeding from the initial state to the final state along the straight line specified in Prob. $7.10$ and back to the initial state along the adiabatic curve. Calculate:(a) The work done on the gas during the adiabatic process.(b) The net work done in the cycle.(c) The net heat transferred to the gas.(d) The thermal efficiency of the cycle.(e) The thermal efficiency of a Carnot cycle operating between a reservoir at the maximum temperature in the cycle and a reservoir at the minimum temperature in the cycle.
A logarithmic thermodynamic temperature scale, which can be constructed to agree with the Celsius scale at $\mathrm{NMP}-\mathrm{H}_{2} \mathrm{O}$ and $\mathrm{NBP} \cdot \mathrm{H}_{2} \mathrm{O}$, is related to the Celsius scale by the formula$$L=(99.974) \frac{\log T-\log 273.15}{\log 373.124-\log 273.15}$$where $L$ is any temperature on the logarithmic temperature scale and $T$ is the corresponding Kelvin temperature on the linear temperature scale. The formula reduces to$$L=738.08 \log T-1798.26$$(a) Calculate logarithmic temperatures for several representative temperatures between $10^{-3} \mathrm{~K}$ and $15 \times 10^{6} \mathrm{~K}$(b) Consider two different temperature ranges on the linear scale that yield the same thermal efficiency for a Carnot cycle. Calculate the temperature ranges on the logarithmic scale and draw a conclusion about efficiency on the logarithmic scale.