1. Find the total differential of \(G\). As always, show your work.
2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

Consider a system which has an internal energy \(U\) defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal energy is an extensive quantity. What constraint does this place on the values \(\alpha\) and \(\beta\) may have?

The internal energy of helium gas at temperature \(T\) is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

1. Compute the entropy.

2. Work out the heat capacity at constant pressure \(C_p\).

3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

4. Compute the internal energy \(U\).

Find the differential of each of the following expressions; zap each of the following with \(d\):

1. \[f=3x-5z^2+2xy\]

2. \[g=\frac{c^{1/2}b}{a^2}\]

3. \[h=\sin^2(\omega t)\]

4. \[j=a^x\]

5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.^*

1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

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.

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 \(U(x,y)\).

1. Find the energy eigenvalues and energy eigenstates for the system.
2. Find \(|\psi(t)\rangle\).
3. List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
4. List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.

The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

Consider the diagram of \(T\) vs \(V\) for several different constant values of \(p\).

1. Translate this diagram to a \(p\) vs \(V\) w/ constant \(T\) graph, including the point \(A\). Complete your graph by hand and make a fairly accurate sketch by printing out the attached grid or in some other way making nice square axes with appropriate tick marks.

   

2. Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?

3. Sketch the line of constant temperature that passes through the point \(A\).

4. What are the values of all the thermodynamic variables associated with the point A?

Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

1. Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

2. Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

For electrons with an energy \(\varepsilon\gg mc^2\), where \(m\) is the mass of the electron, the energy is given by \(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons in a cube of volume \(V=L^3\) the momentum takes the same values as for a non-relativistic particle in a box.

1. Show that in this extreme relativistic limit the Fermi energy of a gas of \(N\) electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where \(n\equiv \frac{N}{V}\) is the number density.

2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}