Physics 4311: Thermal Physics – Homework 7

due date: Wednesday, Mar 18, 2026

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Problem 1: Heat pump (6 points)

A heat pump is a device that uses work to transport heat from a low-temperature reservoir to a high-temperature reservoir (see figure on the right). Its efficiency can be defined as the ratio of the heat deposited into the high-temperature reservoir and the work done on the system, $η = | Q_{h} | ∕ W$ .

a): Heat pumps can be used to heat buildings. What is the efficiency of an ideal (Carnot) heat pump that takes heat from the outside air at 40 $^{\circ}$ F and transports it to the inside of the building which is at 72 $^{\circ}$ F?
b): What is the efficiency of the heat pump in part c) if the outside air is at a temperature of 0 $^{\circ}$ F?

Problem 2: Air conditioner (14 points)

An ideal air conditioner consists of a Carnot cycle (running backwards). It absorbs heat from the inside of a house at the lower temperature $T_{l}$ and discharges heat to the outside at the higher temperature $T_{h}$ , consuming electric energy $E$ . The heat leaking back into the house through the walls and windows is given by $Δ Q = A (T_{h} - T_{l})$ where $A$ is a constant.

a): The air conditioner runs continuously, and the temperature in the house has reached a steady state. Derive a relation for the inside temperature $T_{l}$ in terms of $T_{h}$ , $A$ , and $E$ .
b): The system is designed such that it runs at half of the maximum electrical energy input if the outside temperature is 90 $^{\circ}$ F and the inside temperature is 72 $^{\circ}$ F. What is the highest outside temperature for which the system can maintain an inside temperature of 72 $^{\circ}$ F at full electrical input.

Problem 3: Carnot process for a paramagnetic substance (20 points)

Consider a paramagnetic substance whose equation of state reads $M = αH ∕ T$ where $T$ is the temperature, $M$ is the magnetization, $H$ is the magnetic field, and $α$ is a material specific constant. The internal energy is given by $U = CT$ where the specific heat $C$ is a constant. The work differential for a paramagnet is $δW = HdM$

a): Consider an isothermal change of magnetization from $M_{1}$ to $M_{2}$ . Compute the work $W_{12}$ done on the system and the heat $Q_{12}$ absorbed by the system.
b): Consider an adiabatic change of magnetization from $M_{2}$ to $M_{3}$ . Find the adiabatic $H - M$ curves by starting from $δQ = 0$ and integrating the resulting differential equation.
c): Sketch a Carnot cycle, consisting of two isothermal changes in $M$ and two adiabatic changes in $M$ in the $M - H$ plane.
d): Compute the total work during the cycle and the heat absorbed during the four segments of the cycle.
e): Explicitly calculate the efficiency.

Hint: The derivation is analogous to that of the Carnot cycle for the ideal gas, but using the equation of state $M = αH ∕ T$ instead of the ideal gas law. This leads to changes in some of the mathematical expressions.