## Heat pump

• Carnot efficiency Work Entropy Heat pump
• assignment Light bulb in a refrigerator

assignment Homework

##### Light bulb in a refrigerator
Carnot refridgerator Work Entropy Thermal and Statistical Physics 2020 A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?
• face Work, Heat, and cycles

face Lecture

120 min.

##### Work, Heat, and cycles
Thermal and Statistical Physics 2020

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
• assignment Photon carnot engine

assignment Homework

##### Photon carnot engine
Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature $T$ is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

1. Given $T_H$ and $T_C$, as well as $V_1$ and $V_2$ (the two volumes at $T_H$), determine $V_3$ and $V_4$ (the two volumes at $T_C$).

2. What is the heat $Q_H$ taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at $T_H$ and show that the energy conversion efficiency is the Carnot efficiency.

• assignment Power Plant on a River

assignment Homework

##### Power Plant on a River
efficiency heat engine carnot Energy and Entropy 2021 (2 years)

At a power plant that produces 1 GW ($10^{9} \text{watts}$) of electricity, the steam turbines take in steam at a temperature of $500^{o}C$, and the waste energy is expelled into the environment at $20^{o}C$.

1. What is the maximum possible efficiency of this plant?

2. Suppose you arrange the power plant to expel its waste energy into a chilly mountain river at $15^oC$. Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 10 cents per kilowatt-hour?

3. At what rate will the plant expel waste energy into this river?

4. Assume the river's flow rate is 100 m$^{3}/$s. By how much will the temperature of the river increase?

5. To avoid this “thermal pollution” of the river the plant could instead be cooled by evaporation of river water. This is more expensive, but it is environmentally preferable. At what rate must the water evaporate? What fraction of the river must be evaporated?

• assignment Heat shields

assignment Homework

##### Heat shields
Stefan-Boltzmann blackbody radiation Thermal and Statistical Physics 2020 A black (nonreflective) sheet of metal at high temperature $T_h$ is parallel to a cold black sheet of metal at temperature $T_c$. Each sheet has an area $A$ which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
1. Solve for the net power transferred between the two sheets.

2. A third black metal sheet is inserted between the other two and is allowed to come to a steady state temperature $T_m$. Find the temperature of the middle sheet, and solve for the new net power transferred between the hot and cold sheets. This is the principle of the heat shield, and is part of how the James Web telescope shield works.
3. Optional: Find the power through an $N$-layer sandwich.

• face Review of Thermal Physics

face Lecture

30 min.

##### Review of Thermal Physics
Thermal and Statistical Physics 2020

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
• face Basics of heat engines

face Lecture

10 min.

##### Basics of heat engines
Contemporary Challenges 2022 (3 years) This brief lecture covers the basics of heat engines.
• assignment Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is $22^\circ$C at the ocean surface and $4^{o}$C at the ocean floor.

1. What is the maximum possible efficiency of an engine operating between these two temperatures?

2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water $c_p = 4.2$ Jg$^{-1}$K$^{-1}$ and the density of water is 1 g cm$^{-3}$, and both are roughly constant over this temperature range.

• assignment Active transport

assignment Homework

##### Active transport
Active transport Concentration Chemical potential Thermal and Statistical Physics 2020

The concentration of potassium $\text{K}^+$ ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of $10^4$ the concentration of $\text{K}^+$ ions in the pond water in which the cell is growing. The chemical potential of the $\text{K}^+$ ions is higher in the sap because their concentration $n$ is higher there. Estimate the difference in chemical potential at $300\text{K}$ and show that it is equivalent to a voltage of $0.24\text{V}$ across the cell wall. Take $\mu$ as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

David adds
You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

• assignment Vectors

assignment Homework

##### Vectors
vector geometry Static Fields 2022 (3 years)

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than $\pi/2$ between them?
4. Have an angle of more than $\pi/2$ between them?

• Thermal and Statistical Physics 2020
1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature $T_{HH}$ and the cold (outdoors) temperature $T_C$. What is the ratio $\frac{Q_{HH}}{Q_H}$ of the heat consumed at $T_{HH}$ (i.e. fuel burned) to the heat delivered at $T_H$ (in the house we want to heat)? Give numerical values for $T_{HH}=600\text{K}$; $T_{H}=300\text{K}$; $T_{C}=270\text{K}$.

3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.