Chapter 3, The Second Law and Heat Engines Video Solutions, Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Problem 1

Besides the heat engine and refrigerator, there is a third common appliance known as the heat pump. The main objective of a heat pump is to deliver heat to a high-temperature system. It operates by receiving a certain amount of external work $W=Q_H-Q_L$ to withdraw heat $Q_L$ from a low-temperature reservoir of temperature $T_L$ and depositing heat $Q_H$ into the high-temperature reservoir (the system) of temperature $T_H>T_L$. Suggest a measure for the performance of a heat pump (call it the coefficient of performance $C O P_{H P}$ ) and express the maximum $C O P_{H P}$ in terms of $T_L$ and $T_H$.
As a concrete example, a building at temperature $T$ is heated by a heat pump which uses a river at temperature $T_0$ as a heat source. The heat pump, which has the ideal performance, consumes a constant power $W$ while the building loses heat to its surroundings at a rate $\alpha\left(T-T_0\right)$, where $\alpha$ is a constant. Show that the equilibrium temperature of the building, $T_e>T_0$, is given by
$$
T_e=T_0+\frac{W}{2 \alpha}\left(1+\sqrt{1+\frac{4 \alpha T_0}{W}}\right) .
$$

Vipender Yadav

Numerade Educator

02:43

Problem 2

A Brayton cycle uses an ideal gas as its working substance and consists of the four following reversible steps. The ideal gas is first compressed adiabatically and then expanded isobarically. Afterwards, it is further expanded adiabatically and finally compressed isobarically to its initial state. Determine the efficiency of this engine in terms of the temperatures of the first and second states, $T_1$ and $T_2$.

Ajay Singhal

Numerade Educator

16:52

Problem 3

An Otto cycle uses an ideal gas with an adiabatic index $\gamma$ as its working substance and consists of the four following reversible steps. The ideal gas is first compressed adiabatically and its pressure is then increased isochorically. Afterwards, it is expanded adiabatically and its pressure is finally decreased isochorically to its initial value. Determine the efficiency of this engine in terms of the volumes of the first and second states, $V_1$ and $V_2$, and $\gamma$.

Sheh Lit Chang

University of Washington

10:01

Problem 4

A Stirling cycle uses an ideal gas with an adiabatic index $\gamma$ as its working substance and consists of the four following reversible steps. The ideal gas is first expanded isothermally and its pressure is then decreased isochorically. Afterwards, it is compressed isothermally and its pressure is isochorically increased to its initial value. Determine the efficiency of this engine in terms of the volumes of the first and third states and $\gamma$.

Khoobchandra Agrawal

Numerade Educator

12:12

Problem 5

We have shown that the Carnot engine is the most efficient engine when operating between two heat reservoirs with temperatures $T_H$ and $T_L\left(T_H>\right.$ $T_L$ ), whereby its efficiency is given by $\eta_{\text {Carnot }}=1-\frac{T_L}{T_H}$. Now, suppose that you are given a series of reservoirs with temperatures ranging between $T_L$ and $T_H$ and are asked to construct a heat engine that interacts with any subset of these reservoirs. Show that the efficiency of an engine with internal irreversibilities (e.g. friction) but a well-defined temperature at every juncture (i.e. its process is quasistatic such that it is always in an equilibrium state) is smaller than that of an internally reversible engine following the same cycle of equilibrium states (hint: use Clausius' inequality). Next, prove that the efficiency of an internally reversible engine constructed with the given reservoirs is no larger than the Carnot efficiency $\eta_{\text {Carnot }}=1-\frac{T_L}{T_H}$. This shows that a Carnot engine operating between the highest and lowest temperature reservoirs is still the most efficient when a series of reservoirs with intermediate temperatures is available.

Eric Mockensturm

Numerade Educator

05:21

Problem 6

Determine the entropy change of $n$ moles of a gas with an adiabatic constant $\gamma$ in expanding from an initial volume $V$ to a final volume $k V$ under isothermal and isobaric conditions.

Salamat Ali

Numerade Educator

02:22

Problem 7

A thermally insulated container of total volume $V$ is separated by a frictionless divider into two compartments $\mathrm{A}$ and $\mathrm{B}$ that have volumes $\alpha V$ and $(1-\alpha) V$ respectively. $n$ moles of a certain gas fills compartment $A$ and a certain amount of a different gas fills compartment $B$ such that the system is in equilibrium. Determine the entropy change, of the system comprising the two gases, associated with mixing the two gases by removing the divider and waiting till the system attains thermodynamic equilibrium once again.

Ajay Singhal

Numerade Educator

01:39

Problem 8

Two thermally insulated vessels of volumes $V_1$ and $V_2$ initially contain $n_1$ and $n_2$ moles of different monoatomic gases that are at pressures $p_1$ and $p_2$. These vessels are then connected by a thermally insulated tube. After the system of vessels equilibrates, determine the change in entropy of the universe.

Dominador Tan

Numerade Educator

01:04

Problem 9

A small body with constant heat capacity $C$ is placed in direct thermal contact with a large reservoir of temperature $T_2$ such that its temperature is changed from $T_1$ to $T_2$. Determine the entropy changes of the body, reservoir and the universe. Show that the entropy change of the universe is nonnegative regardless of the relative magnitudes of $T_1$ and $T_2$. Now if the heat is delivered to or extracted from the small body via a Carnot engine operating between the large reservoir and the small body, determine the entropy changes of the body, reservoir and the universe.

Narayan Hari

Numerade Educator

03:03

Problem 10

Two substances of heat capacities $C_1, C_2$ and initial temperatures $T_1$ and $T_2$ are placed in thermal contact. They are isolated from their surroundings. When thermal equilibrium is subsequently achieved, determine the entropy change of the universe and show that it must be non-negative.

Satpal Satpal

Numerade Educator

02:25

Problem 11

Determine the maximum work obtainable from a heat engine connected to two reservoirs of constant heat capacities $C_H$ and $C_L$ at initial temperatures $T_H$ and $T_L<T_H$.

Penny Riley

Numerade Educator

22:38

Problem 12

A cylindrical container is separated by a fixed divider with a valve and a frictionless piston is attached to its open right end. The walls of the cylinder, divider and piston are perfect thermal insulators. The cylinder is filled with $12 \mathrm{~g}$ of helium in the left compartment and $2 \mathrm{~g}$ of helium in the right. Initially, the pressures, volumes and temperatures of the gases are respectively, $6 \mathrm{~atm}$,$11.2 \mathrm{~L}$ and $273 \mathrm{~K}$ in the left side, and $1 \mathrm{~atm}, 11.2 \mathrm{~L}$ and $273 \mathrm{~K}$ in the right side. The specific heat capacity (note that this is per unit mass) of helium at constant pressure is $c_p=5.25 \mathrm{~J} / \mathrm{g} \mathrm{K}$. The piston is pushed towards the divider by a reversible compression until the pressure on the right side equals $6 \mathrm{~atm}$. At this juncture, the valve opens and the whole system is allowed to reach equilibrium. What is the final equilibrium temperature? Find the total entropy change of the whole process. (Singapore Physics Olympiad)

Brandy Heflin

Numerade Educator

01:04

Problem 13

A body of constant heat capacity $C$ is heated up from a temperature $T_1$ to $T_2$ by bringing it into thermal contact and waiting till it establishes thermal equilibrium with $N$ large reservoirs of temperatures $T_1+\Delta T$, $T_1+2 \Delta T, \ldots, T_1+(N-1) \Delta T, T_2$ in ascending order of temperature, where $\Delta T=\frac{T_2-T_1}{N}$. Determine the net entropy change of the universe due to this process. Then, take the limit of $N \rightarrow \infty$ and $\Delta T \rightarrow d T$ where $d T$ is an infinitesimal change in temperature and show that this process is reversible.

Narayan Hari

Numerade Educator