Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.
Write a thermodynamic derivative on the board, and ask the students to describe the experiment that you would perform in order to measure it, and draw a picture of the apparatus.
Your group will be given one of the following partial derivatives: \begin{align} a)\quad\left(\frac{\partial {V}}{\partial {p}}\right)_{T}\quad b)\quad\left(\frac{\partial {U}}{\partial {p}}\right)_{S}\quad c)\quad\left(\frac{\partial {T}}{\partial {V}}\right)_{S}\quad d)\quad\left(\frac{\partial {V}}{\partial {T}}\right)_{p}\quad e)\quad\left(\frac{\partial {U}}{\partial {T}}\right)_{V}\\ f)\quad\left(\frac{\partial {p}}{\partial {V}}\right)_{T}\quad g)\quad\left(\frac{\partial {V}}{\partial {T}}\right)_{S}\quad h)\quad\left(\frac{\partial {T}}{\partial {V}}\right)_{p}\quad i)\quad\left(\frac{\partial {T}}{\partial {U}}\right)_{V}\quad j)\quad\left(\frac{\partial {V}}{\partial {p}}\right)_{S} \end{align} In your group, design an experiment to measure this derivative. Draw a sketch of the apparatus and describe how to convert directly measured data into a numerical value for the derivative.
If you finish with your derivative, you can try designing an experiment for the next derivative in the list.
Partials that should be considered for this activity:
A particularly challenging pair of derivatives are \(\left(\frac{\partial p}{\partial S}\right)_T\) and \(\left(\frac{\partial V}{\partial S}\right)_T\). In particular, the idea of “heating” something at constant temperature is quite counterintuitive. It may help to invoke the example of melting ice, in which you are heating the ice, but it stays at zero centigrade.
If many or most of the groups had trouble with a particular concept, it's worth bringing everyone together to discuss this. As well, if there was a particular group that had a unique solution, it is worth showing to the class as well. Here is a [[whitepapers:narratives:intro|narrative]] for this activity.
group Small Group Activity
30 min.
face Lecture
5 min.
thermodynamics statistical mechanics
This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.group Small Group Activity
30 min.
assignment Homework
Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call \(y\), and the distance of horizontal stretch we will call \(x\).
If I pull the bottom down by a small distance \(\Delta y\), with no horizontal force, what is the resulting change in width \(\Delta x\)? Express your answer in terms of partial derivatives of the potential energy \(U(x,y)\).
assignment Homework
In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.
Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)
In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.^{*}
Thus your savings \(S\) can be considered to be a function of your bagels \(B\) and coffee \(C\). In this problem we will also discuss the prices \(P_B\) and \(P_C\), which you may not assume are independent of \(B\) and \(C\). It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.
The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of \(P_C\) and \(P_B\)?
Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).
assignment Homework
Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).
Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).
group Small Group Activity
30 min.
assignment Homework
Find the entropy of a set of \(N\) oscillators of frequency \(\omega\) as a function of the total quantum number \(n\). Use the multiplicity function: \begin{equation} g(N,n) = \frac{(N+n-1)!}{n!(N-1)!} \end{equation} and assume that \(N\gg 1\). This means you can make the Sitrling approximation that \(\log N! \approx N\log N - N\). It also means that \(N-1 \approx N\).
Let \(U\) denote the total energy \(n\hbar\omega\) of the oscillators. Express the entropy as \(S(U,N)\). Show that the total energy at temperature \(T\) is \begin{equation} U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \end{equation} This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.