assignment Homework

Gibbs free energy
thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
  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 \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

assignment Homework

Extensive Internal Energy
Energy and Entropy 2021 (2 years)

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?

assignment Homework

Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy 2021 (2 years)

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}

Consider a very irreversible process in which a small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?

assignment Homework

Using Gibbs Free Energy
thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

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 \(U\).

assignment Homework

Zapping With d 1
Energy and Entropy 2021 (2 years)

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

face Lecture

30 min.

A coarse-grained model for transportation
Contemporary Challenges 2022 (4 years)

energy flow diagram energy car

A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

assignment Homework

Free Expansion
Energy and Entropy 2021 (2 years)

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.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
  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?

assignment Homework

Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

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.

format_list_numbered Sequence

Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.

assignment Homework

Rubber Sheet
Energy and Entropy 2021 (2 years)

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

30 min.

Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

blackbody radiation

This lecture is one step in motivating the form of the Planck distribution.

assignment Homework

Isothermal/Adiabatic Compressibility
Energy and Entropy 2021 (2 years)

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}

assignment Homework

Translating Contours
Energy and Entropy 2021 (2 years)

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?

assignment Homework

Distribution function for double occupancy statistics
Orbitals Distribution function Thermal and Statistical Physics 2020

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)?

assignment Homework

Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass \(M\) at temperature \(T\) in a uniform gravitational field \(g\). Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom \(h=0\) of the column. Integrate from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.

group Small Group Activity

30 min.

Mass is not Conserved
Theoretical Mechanics (4 years)

energy conservation mass conservation collision

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

group Small Group Activity

30 min.

Hydrogen emission
Contemporary Challenges 2022 (5 years)

hydrogen atom photon energy

In this activity students work out energy level transitions in hydrogen that lead to visible light.

group Small Group Activity

30 min.

Grey space capsule
Contemporary Challenges 2022 (4 years)

blackbody Stefan-Boltzmann Law

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

assignment Homework

Energy of a relativistic Fermi gas
Fermi gas Relativity Thermal and Statistical Physics 2020

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}

group Small Group Activity

10 min.

Thermal radiation at twice the temperature
Contemporary Challenges 2022 (4 years)

Stefan-Boltzmann blackbody radiation

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.