## Activity: Applying the equipartition theorem

Contemporary Challenges 2022 (4 years)
Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• Media
This follows Equipartition theorem
If the microscopic world was classical, predict $U_{\text{classical}}(T)$ for the following “toy molecules” in the gas phase.
• Each ball is a point mass $m$ with no moment of inertia.
• The zig-zag lines are springs which are freely jointed at the balls.
• Vibrational motion of the springs is very small ($\ll$ the length of the spring).
• The springs can extend and compress, but cannot twist or flex.
• The straight lines are rigid rods.
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.
• 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 Basic Calculus: Practice Exercises

assignment Homework

##### Basic Calculus: Practice Exercises
Static Fields 2022 (3 years) Determine the following derivatives and evaluate the following integrals.
1. $\frac{d}{du}\left(u^2\sin u\right)$
2. $\frac{d}{dz}\left(\ln(z^2+1)\right)$
3. $\displaystyle\int v\cos(v^2)\,dv$
4. $\displaystyle\int v\cos v\,dv$
• assignment Einstein condensation temperature

assignment Homework

##### Einstein condensation temperature
Einstein condensation Density Thermal and Statistical Physics 2020

Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.

Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.

• group Grey space capsule

group Small Group Activity

30 min.

##### Grey space capsule
Contemporary Challenges 2022 (4 years)

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.
• assignment One-dimensional gas

assignment Homework

##### One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of $N$ particles, each of mass $M$, confined to a one-dimensional line of length $L$. The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature $T$. You may assume that the temperature is high enough that $k_B T$ is much greater than the ground state energy of one particle.
• face Fermi and Bose gases

face Lecture

120 min.

##### Fermi and Bose gases
Thermal and Statistical Physics 2020

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
• assignment Free energy of a two state system

assignment Homework

##### Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

• group de Broglie wavelength after freefall

group Small Group Activity

30 min.

##### de Broglie wavelength after freefall
Contemporary Challenges 2022 (4 years)

In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.
• face Phase transformations

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

Learning Outcomes