Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
Little introduction is necessary for this activity; however, the first Name the experiment activity should be performed before entering this Name the Experiment activity. Be sure to state to the groups that they must measure the partial derivatives they are given for a rubber band, providing both a description and a picture of their experiment.
Your group will be given one of the following partial derivatives: \begin{align} a)\quad\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\quad b)\quad\left(\frac{\partial {T}}{\partial {S}}\right)_{p}\quad c)\quad\left(\frac{\partial {S}}{\partial {V}}\right)_{T}\quad d)\quad\left(\frac{\partial {p}}{\partial {S}}\right)_{V}\quad e)\quad\left(\frac{\partial {S}}{\partial {p}}\right)_{T}\\ f)\quad\left(\frac{\partial {p}}{\partial {S}}\right)_{T}\quad g)\quad\left(\frac{\partial {T}}{\partial {S}}\right)_{V}\quad h)\quad\left(\frac{\partial {V}}{\partial {S}}\right)_{T}\quad i)\quad\left(\frac{\partial {V}}{\partial {S}}\right)_{p}\quad j)\quad\left(\frac{\partial {S}}{\partial {T}}\right)_{p} \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.
For this activity we have just four partial derivatives:
If a group finishes their experiment early, have them create an experiment for a more challenging partial derivative.
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.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
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 \(U(x,y)\).
group Small Group Activity
60 min.
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 \(T\rightarrow 0\).
assignment Homework
group Small Group Activity
30 min.
group Small Group Activity
120 min.
Projectile Motion Drag Forces Newton's 2nd Law Separable Differential Equations
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.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.