These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
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.
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}
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 \(T_C\).
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}
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.
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.
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.
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 Homework
assignment Homework
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?
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 \(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\); \(T_{C}=270\text{K}\).
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 Homework
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).
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 \(T_C\)).
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?
Does the work done on the two isentropic stages cancel each other, as for the ideal gas?
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 Homework
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.
What is the maximum possible efficiency of an engine operating between these two temperatures?
face Lecture
30 min.
thermodynamics statistical mechanics
These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.assignment Homework
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 \(20^{o}C\).
What is the maximum possible efficiency of this plant?
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?
At what rate will the plant expel waste energy into this river?
Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will the temperature of the river increase?
face Lecture
10 min.
assignment Homework
Solve for the net power transferred between the two sheets.
Optional: Find the power through an \(N\)-layer sandwich.
assignment Homework
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.
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)?
group Small Group Activity
60 min.
Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics
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.