- Ideal gas Entropy Sackur-Tetrode
*assignment*Entropy, energy, and enthalpy of van der Waals gas*assignment*Homework##### Entropy, energy, and enthalpy of van der Waals gas

Van der Waals gas Enthalpy Entropy Thermal and Statistical Physics 2020In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.

Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

*face*Work, Heat, and cycles*face*Lecture120 min.

##### Work, Heat, and cycles

Thermal and Statistical Physics 2020work heat engines Carnot thermodynamics entropy

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.*face*Ideal Gas*face*Lecture120 min.

##### Ideal Gas

Thermal and Statistical Physics 2020ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.*face*Boltzmann probabilities and Helmholtz*face*Lecture120 min.

##### Boltzmann probabilities and Helmholtz

Thermal and Statistical Physics 2020ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.*assignment*Entropy of mixing*assignment*Homework##### Entropy of mixing

Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

Show that after diffusive equilibrium is reached the total entropy is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is known as the entropy of mixing.

If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

Since the Helmholtz free energy is lower for the mixed \(AB\) than for the separated \(A\) and \(B\), it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

- Note
This course has not yet covered

*work*, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

*assignment*Quantum harmonic oscillator*assignment*Homework##### Quantum harmonic oscillator

Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020Find 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.

*assignment*Vapor pressure equation*assignment*Homework##### Vapor pressure equation

phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is*way*bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase.**Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.**Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.

Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

*group*Hydrogen emission*group*Small Group Activity30 min.

##### Hydrogen emission

Contemporary Challenges 2022 (4 years) In this activity students work out energy level transitions in hydrogen that lead to visible light.*assignment*Centrifuge*assignment*Homework##### Centrifuge

Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\)*on*the axis. Take \(\mu\) as for an ideal gas.*assignment*Quantum concentration*assignment*Homework##### Quantum concentration

bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.-
Thermal and Statistical Physics 2020
Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.

How much heat (in joules) is added to the gas in each of these two processes?

What is the temperature at the end of the second process?

Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?