Student handout: 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.

Week 8:Work, heat, and cycles (K&K 8, Schroeder 4)

This week we will be zooming through chapters 8 of Kittel and Kroemer. Chapter 8 covers heat and work, which you learned about during Energy and Entropy. Hopefully this will be a bit of review and catch-up time, before we move on to phase transitions.

Heat and work

As we reviewd in week 1, heat and work for a quasistatic process are given by \begin{align} Q &= \int TdS \\ W &= -\int pdV \end{align} But we can often make use of the First Law in order to avoid computing both of these (if we know how to find the internal energy): \begin{align} \Delta U &= Q + W \end{align}

Carnot cycle

We have a monatomic ideal gas, and you can use any of its properties that we have worked out in class. We can begin with what you saw in Energy and Entropy \begin{align} pV &= NkT \\ U &= \frac32 NkT \end{align} and we can add to that the results from this class: \begin{align} S&= Nk\left(\ln\left(\frac{n_Q}{n}\right) + \frac52\right) \\ F &= NkT\left(\ln\left(\frac{n}{n_Q}\right) -1\right) \\ n &= n_Q e^{-\beta\mu} \\ n_Q &\equiv\left(\frac{mkT}{2\pi \hbar^2}\right)^{\frac32} \\ \end{align}

Let us consider a simple cycle in which we start with the gas at temperature \(T_C\).

  1. Adiabatically compress the gas until it reaches temperature \(T_H\).
  2. Expand a gas to twice its original volume at fixed temperature \(T_H\).
  3. Expand the gas at fixed entropy until its temperature reaches \(T_C\).
  4. Finally go back to the original volume at fixed temperature \(T_C\).
Small groups
Solve for the heat and work on each of these steps. In addition find the total work done.
Answer
We can solve this problem most easily by working out the heat at each step.
  1. Since the process is adiabatic, \(Q_1=0\). To find the work, we just need to know \(\Delta U=\frac32 Nk\Delta T\). So the work must be \(W = \frac32 Nk\Delta (T_H-T_C)\).
  2. Now we are increasing the volume, which will change the entropy. Since the temperature is fixed, \(Q=T\Delta S\), and we can find \(\Delta S\) easily enough from the Sackur-Tetrode entropy: \(\Delta S = Nk\ln 2\). Since the internal energy doesn't change, the heat and work are opposite. \(Q = -W = NkT_H\ln 2\).
  3. Now we are again not changing the entropy, and thus not heating the system, so \(W =\Delta U\), and the work done is equal and opposite of the work done on step #1. \(W = \frac32 Nk\Delta (T_C-T_H)\).
  4. This will be like step 2, but now the temperature is different, and since we are compressing the work is positive while the heat is negative: \(Q = -W = NkT_C\ln\frac12 = -NkT_C\ln 2\).

Putting these all together, the total work done is \begin{align} W &= NkT_H\ln 2 - NkT_C\ln 2 \\ &= \ln 2Nk (T_H-T_C) \end{align}

Efficiency of an engine

If we are interested in this as a heat engine, we have to ask what we put into it. This diagram shows where energy and entropy go. The engine itself (our ideal gas in this case) returns to its original state after one cycle, so it doesn't have any changes. However, we have a hot place (where the temperature is \(T_H\), which has lost energy due to heating our engine as it expanded in step 2), and a cool place at \(T_C\), which got heated up when we compressed our gas at step 4. In addition, over the entire cycle some work was done.

Carnot engine energy and entropy flow diagram. The entropy lost by the hot place is the same as the entropy gained by the cold place, because the Carnot engine is reversible.

The energy we put in is all the energy needed to keep the hot side hot, which is the \(Q\) for step 2. \begin{align} Q_H &= NkT_H\ln 2 \end{align} The efficiency is the ratio of what we get out to what we put in, which gives us \begin{align} \varepsilon &= \frac{W}{Q_H} \\ &= \frac{\ln 2Nk (T_H-T_C)}{NkT_H\ln 2} \\ &= 1 - \frac{T_C}{T_H} \end{align}' and this is just the famous Carnot efficiency.

Note
I could have made this an easier problem if I had changed the statement to expand at fixed temperature until the entropy changed by a given \(\Delta S\). Then we would not have had to use the Sackur-Tetrode equation at all, and our result would have been true for any material, not just an ideal gas!

We could also have run this whole cycle in reverse. That would look like the next figure. This is how a refridgerator works. If you had an ideal refridgerator and an ideal engine with equal capacity, you could operate them both between the inside and outside of a room to acheive nothing. The engine could precisely power the refridgerator such that no net heat is exchanged between the room and its environment.

Carnot fridge energy and entropy flow diagram.

Naturally, we cannot create an ideal Carnot engine or an ideal Carnot refridgerator, because in practice a truly reversible engine would never move. However, it is also very useful to know these fundamental limits, which can guide real heat engines (e.g. coal or nuclear power plants, some solar power plands) and refridgerators or air conditioners. Another use of this ideal picture is that of a heat pump, which is a refridgerator in which you cool the outside in order to heat your house (or anything else). A heat pump can thus be more efficient than an ordinary heater. Just looking at the diagram for a Carnot fridge, you can see that the heat in the hot location exceeds the work done, preciesly because it also cools down the cold place.
  • 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?
  • assignment Heat pump

    assignment Homework

    Heat pump
    Carnot efficiency Work Entropy Heat pump 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.

  • 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 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.

  • face Review of Thermal Physics

    face Lecture

    30 min.

    Review of Thermal Physics
    Thermal and Statistical Physics 2020

    thermodynamics statistical mechanics

    These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
  • 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?

  • face Basics of heat engines

    face Lecture

    10 min.

    Basics of heat engines
    Contemporary Challenges 2021 (4 years) This brief lecture covers the basics of heat engines.
  • 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.

  • assignment Adiabatic Compression

    assignment Homework

    Adiabatic Compression
    ideal gas internal energy engine Energy and Entropy 2020

    A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

    In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

    Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

    1. If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

    2. By what factor does the pressure increase?

  • group Gravitational Potential Energy

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    60 min.

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    Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
Learning Outcomes