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 \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
  • Found in: Energy and Entropy course(s)

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?

  • Found in: Energy and Entropy course(s)

face Lecture

10 min.

Introduction to Central Forces
  • Found in: Central Forces course(s)
A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

group Small Group Activity

5 min.

Constant Lines in the \(u\), \(v\)-Plane

Students are asked to draw lines of constant \(u\) and \(v\) in a \(u,v\) coordinate system. Then, in the same coordinate system, students must draw lines of constant \(x\) and constant \(y\) when

\[x(u,v)=u \] and \[y(u,v)=\frac{1}{2}u+3v. \]

A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?
  • Found in: Thermal and Statistical Physics course(s)

assignment_ind Small White Board Question

5 min.

Angular Momentum SWBQ
  • Found in: Central Forces course(s)

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?

  • Found in: Energy and Entropy course(s)

group Small Group Activity

10 min.

Angular Momentum in Polar Coordinates
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates

group Small Group Activity

60 min.

Multivariable Pictionary
Students draw the 3D graphs of equations using three variables. They make choices for drawing a stack of curves in parallel planes and a curve in a perpendicular plane (e.g. substituting in values for \(x\), \(y\), or \(z\). )

None

Heat pump
  1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

  2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature \(T_{HH}\) and the cold (outdoors) temperature \(T_C\). What is the ratio \(\frac{Q_{HH}}{Q_H}\) of the heat consumed at \(T_{HH}\) (i.e. fuel burned) to the heat delivered at \(T_H\) (in the house we want to heat)? Give numerical values for \(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\); \(T_{C}=270\text{K}\).

  3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.

  • Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

30 min.

Chain Rule
This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.
  • Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

Work By An Electric Field (Contour Map)
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

face Lecture

120 min.

Work, Heat, and cycles
These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
A black (nonreflective) sheet of metal at high temperature \(T_h\) is parallel to a cold black sheet of metal at temperature \(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
  1. Solve for the net power transferred between the two sheets.

  2. A third black metal sheet is inserted between the other two and is allowed to come to a steady state temperature \(T_m\). Find the temperature of the middle sheet, and solve for the new net power transferred between the hot and cold sheets. This is the principle of the heat shield, and is part of how the James Web telescope shield works.
  3. Optional: Find the power through an \(N\)-layer sandwich.

  • Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

5 min.

Maxima and Minima
This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.
  • Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

None

Centrifuge
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.
  • Found in: Thermal and Statistical Physics course(s)

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 \(U\).

  • Found in: Energy and Entropy course(s)

group Small Group Activity

5 min.

Fourier Transform of a Plane Wave
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)