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### Unit: Math Bits

**Hour 1: Math Bits - Differentials**- Differentials
- Zapping with d
*Surfaces Activity: Covariation in Thermal Systems*

- QUIZ
*Small Group Activity: Exploring the Partial Derivative Machine*- Contour Graphs
*Surfaces Activity: Thermodynamic States*

*Small Group Activity: Exploring the Partial Derivative Machine*- Partial Derivative Machine dictionary

**Hour 4: Math Bits - Energy and Integrals***dU*- Numerical Integration
*Surfaces Activity: Quantifying Change in Thermal Systems*

**Hours 5-6: Math Bits - Derivatives and Chain Rules**- Chain Rules
*Small Group Activity: PDM Derivatives*

- Chain Rule Diagrams

*Surfaces Activity: “Squishability” of Water Vapor*

### Unit: First Law of Thermodynamics

**Hour 8: Heat and Temperature**- Intro to Thermodynamic Properties
*Lab: Ice Lab I*

- Comparing Thermodynamic Properties

**Hour 9: The first law**- QUIZ
- Lecture: Jargon
- Intensive and Extensive
- T, S, V, p, U
- Work, Heat
- The First Law

- Snapping a Rubber Band

- Thermodynamic Identity
- Name the Experiment I
- Heat Capacity

**Hour 11: Heat capacity and latent heat***Lab: Ice Lab II*

- This should go somewhere around here once things get rearranged:
*Heat Capacity of Water Vapor*

**Hour 12: Mechanical cycles**- Elevators
*Small Group Activity: The Elevator Cycle*

**Hour 13: Heat, work, and processes**- Work
- Heat
- Using $p V$ and $T S$ Plots

**Hour 14: A simple cycle**- Analyzing a Simple Curve

**Hour 15: Holiday**

### Unit: Internal Energy

- Quiz
- Name the Experiment II (Heat and/or First Law)

**Hour 17: The second law**- Lecture: Jargon
- Second Law
- Reversible and Irreversible
- Adiabatic
- Quasistatic

- Free expansion not-quiz
- Discussion of not-quiz

**Hours 19-20: Legendre Transformations**

- (GibbsFreeEnergy)
*What goes here?*Compute the entropy.

Work out the heat capacity at constant pressure $C_{p}$.

Find the connection among $V, p, N,$ and $T$, which is called the equation of state.

Compute the internal energy $U$.

**Hour 21: Maxwell relations**- Maxwell Relations
- Maxwell Relation Activities

- (HardSphereFluid)
*What goes here?***Hard-sphere fluid**A hard-sphere fluid (or gas) is just like and ideal gas except that the gas (fluid) particles are not point particles, but instead have a finite volume. The following equations approximately describe the hard-sphere fluid.\[U=\frac{3}{2} N k_B T\]

\[\eta=\frac{N}{V} \frac{4 \pi}{3} R^3\]

\[\frac{F}{N k_B T}=\ln \left( \frac{N}{V} \left( \frac{3 h^2}{4 \pi m} \frac{N}{U} \right)^\frac32 \right) + \frac{4 \eta - 3 \eta^2}{\left( 1-\eta \right)^2}-\frac12\]

where $k_B$ is a constant, and you may consider the number of spheres $N$ as a constant for this problem. Similarly the mass $m$, the radius $R$, and Plank's constant $h$ are constants. In these equations $\eta$ is called the packing fraction, and represents the fraction of the volume that is occupied by hard spheres (which has no impact whatsoever on how you solve this problem).

Solve for \[S \equiv -\left( \frac{\partial F}{\partial T} \right) _V\]

Solve for \[S \equiv -\left( \frac{\partial F}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _S\]

- (SimpleCycleII)
*What goes here?*

- Name The Experiment III

- (IsothermalAdiabaticCompressibility)
*What goes here?*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}

- (NonIdealGas)
*What goes here?*The equation of state of a gas that departs from ideality can be approximated by \[ p=\frac{NkT}{V}\left(1+\frac{NB_{2}(T)}{V}\right) \] where $B_{2}(T)$ is called the second virial coefficient which increases monotonically with temperature. Find $\left( \frac{\partial U}{\partial V}\right)_{T}$ and determine its sign.

**Hours 23-24: Using a Maxwell relation to make an entropy change**- QUIZ
*Lab: Rubber Band Lab*

- Sometimes Always Never True (1st half)

**Hours 26-28: Holiday**

### Unit: Statistical Mechanics

**Hour 29: Statistical view of entropy**- QUIZ
- Fairness Function and Probability

- (ThermodynamicPotentialsAndMaxwellRelations) PRACTICE
*What goes here?*For the three thermodynamic potentials defined as

\begin{quote} \(F=U-T S\) \hfill Helmholtz free energy

\(H=U+P V\) \hfill Enthalpy

\(U-T S+P V\) \hfill Gibbs free energy \end{quote}

determine the total differentials for each thermodynamic potential: $dF$, $dH$, and $dG$. Use the thermodynamic identity ($dU=T dS-p dV$) to simplify.

Identify a Maxwell relation for each of the three potentials.

- (TwoLevelSystem)
*What goes here?***A two level system**A system consists of $N$ identical non-interacting (independent) objects. Each object has two energy eigenstates labeled $A$ and $B$, with energies $E_A=E_0$ and $E_B=-E_0$ ($E_0=\left|E_0 \right|$).Two students argue that this object can't be real because the low energy state is not zero, i.e. $E_B=0$ and $E_A=2 E_0$. What is the effect of the energy of the lowest energy state on thermal averages?

As a function of temperature $T$, what are the probabilities $p_A$ and $p_B$ of finding an object in this system in eigenstates $A$ and $B$?

Calculate the internal energy $U$ of this system as a function of temperature.

Each object has a property $X$, which takes the values of $X_A=+1$ when an object is in state $A$ and $X_B=-1$ when an object is in state $B$. Solve for the average value $\langle X \rangle$ as a function of temperature of the property $X$ for this system.

What are the limits for $T\rightarrow 0$ and $T\rightarrow\inf$ for $\langle X \rangle$?

**Hour 30: Maximizing Entropy**- Maximizing S
- Lagrange Multipliers

- Weighted Averages
- Derive Boltzman Ratio and connect to Thermo ($U = \sum p_iE_i$)

- (BoltzmannRatio)
*What goes here?*At low temperatures, a diatomic molecule can be well described as a

*rigid rotor*. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{lm} &= \hbar^2 \frac{l(l+1)}{2I} \end{align}What is the energy of the ground state and the first and second excited states of the $H_2$ molecule?

At room temperature, what is the relative probability of finding a hydrogen molecule in the $l=0$ state versus finding it in any one of the $l=1$ states?\\ i.e. what is $P_{l=0,m=0}/\left(P_{l=1,m=-1} + P_{l=1,m=0} + P_{l=1,m=1}\right)$

At what temperature is the value of this ratio 1?

At room temperature, what is the probability of finding a hydrogen molecule in any one of the $l=2$ states versus that of finding it in the ground state?\\ i.e. what is $P_{l=0,m=0}/\left(P_{l=2,m=-2} + P_{l=2,m=-1} + \cdots + P_{l=2,m=2}\right)$

- (NucleusInMagneticField)
*What goes here?*Nuclei of a particular isotope species contained in a crystal have spin $I=1$, and thus, $m = \{+1,0,-1\}$. The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, $E=\varepsilon$, in the state $m=+1$ and the state $m=-1$, compared with an energy $E=0$ in the state $m=0$, i.e. each nucleus can be in one of 3 states, two of which have energy $E=\varepsilon$ and one has energy $E=0$.

Find the Helmholtz free energy $F = U-TS$ for a crystal containing $N$ nuclei which do not interact with each other.

Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

**Hour 32: Free**- Free

**Hours 33-34: Exploring a Diatomic Gas**- Diatomic Gas

**Hour 35: Review**- Review

- (HotMetal)
*No HW problem by this name on archive.*