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### Unit: Math Bits

• 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
• dU
• Numerical Integration
• Surfaces Activity: Quantifying Change in Thermal Systems
• Chain Rules
• Small Group Activity: PDM Derivatives
• Chain Rule Diagrams
• Surfaces Activity: “Squishability” of Water Vapor

### Unit: First Law of Thermodynamics

• 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
• Lab: Ice Lab II
This should go somewhere around here once things get rearranged:
• Heat Capacity of Water Vapor
• Elevators
• Small Group Activity: The Elevator Cycle
• 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
• Quasistatic
• Free expansion not-quiz
• Discussion of not-quiz
1. (GibbsFreeEnergy) What goes here?

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.

4. Compute the internal energy $U$.

• Maxwell Relations
• Maxwell Relation Activities
1. (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).

1. Solve for $S \equiv -\left( \frac{\partial F}{\partial T} \right) _V$

2. Solve for $S \equiv -\left( \frac{\partial F}{\partial V} \right) _T$

3. Solve for $\left( \frac{\partial U}{\partial V} \right) _T$

4. Solve for $\left( \frac{\partial U}{\partial V} \right) _S$

1. (SimpleCycleII) What goes here?

• Name The Experiment III

The isothermal compressibility is defined as $$K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ $K_T$ 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 $$K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}$$ 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 $$\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}$$ 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}

1. (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.

• QUIZ
• Lab: Rubber Band Lab
• Sometimes Always Never True (1st half)
Hours 26-28: Holiday

### Unit: Statistical Mechanics

• QUIZ
• Fairness Function and Probability
1. (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}

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

2. Identify a Maxwell relation for each of the three potentials.

1. (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 $E_B=-E_0$ ($E_0=\left|E_0 \right|$).

1. 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?

2. 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 $B$?

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

4. 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.

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

• Maximizing S
• Lagrange Multipliers
• Weighted Averages
• Derive Boltzman Ratio and connect to Thermo ($U = \sum p_iE_i$)
1. (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}

1. What is the energy of the ground state and the first and second excited states of the $H_2$ molecule?

2. 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 $P_{l=0,m=0}/\left(P_{l=1,m=-1} + P_{l=1,m=0} + P_{l=1,m=1}\right)$

3. At what temperature is the value of this ratio 1?

4. 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 $P_{l=0,m=0}/\left(P_{l=2,m=-2} + P_{l=2,m=-1} + \cdots + P_{l=2,m=2}\right)$

1. (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 $E=0$.

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

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

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

Hour 32: Free
• Free
• Diatomic Gas
Hour 35: Review
• Review

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

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