## Student handout: Review of Thermal Physics

Thermal and Statistical Physics 2020
These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.

Topics are everything that has been covered on homework. Problems should be similar to homework problems, but short enough to be completed during the exam. The exam will be closed notes. You should be able to remember the fundamental equations.

### Equations to remember

Most of the equations I expect you to remember date back from Energy and Entropy, with a few exceptions.

Thermodynamic identity
The thermodynamic identity, including the chemical potential: \begin{align} dU &= TdS - pdV + \mu dN \end{align} You should be able from this to extract relationships such as $\mu = \left(\frac{\partial U}{\partial N}\right)_{S,V}$.
Thermodynamic potentials
You need to know the Helmholtz and Gibbs free energies. \begin{align} F &= U - TS \\ G &= U - TS +pV \\ dF &= -SdT - pdV + \mu dN \\ dG &= -SdT + Vdp + \mu dN \end{align} You don't need to remember their differentials, but you do need to be able to find them quickly and use them, e.g. to find out how $\mu$ relates to $F$ as a derivative. I'll point out that by remembering how to find the differentials, you also don't need to remember the sign of $U-TS$, since you can figure it out from the thermodynamic identity by making the $TdS$ term cancel.
Heat and work
You should remember the expressions for differential heat and work \begin{align} dQ &= TdS \\ dW &= -pdV \end{align} and you should be able to use these expressions fluently, including integrating to find total heat or work, or solving for entropy given heat: \begin{align} dS &= \frac{dQ}{T} \end{align}
Efficiency
You should know that efficiency is defined as “what you get out” divided by “what you put in”, and that for a heat engine this comes down to \begin{align} \epsilon &= \frac{W_{\text{net}}}{Q_H} \end{align}
Entropy
You should remember the Gibbs expression for entropy in terms of probability. \begin{align} S &= -k\sum_i P_i\ln P_i \end{align}
Boltzmann probability
You should be comfortable with the Boltzmann probability, able to predict properties of systems using them. \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} \\ Z &= \sum_i e^{-\beta E_i} \\ F &= -kT\ln Z \end{align}
Derivative trick
You may need to remember the derivative trick for turning a summation into a derivative of another summation in order to complete a problem. More particularly, I want you not to use an expression for $U$ in terms of $Z$ that comes from the derivative trick, without writing down the three lines of math (or so) required to show that it is true.
Thermal averages
You should remember that the internal energy is given by a weighted average: \begin{align} U &= \sum_i E_i P_i \end{align} And similarly for other variables, such as $N$ in the grand canonical ensemble.
Chemical potential
You should remember that the chemical potential is the Gibbs free energy per particle. \begin{align} \mu &= \frac{G}{N} \end{align} You should also be able to make a distinction between internal and external chemical potential to solve problems such as finding the density as a function of altitude (or in a centrifuge), if I give you the expression for the chemical potential of an ideal gas (or other fluid).
Gibbs factor and sum
You should be comfortable with the Gibbs sum and finding probabilities in the grand canonical ensemble. \begin{align} P_i &= \frac{e^{-\beta (E_i-\mu N_i)}}{Z} \\ \mathcal{Z} &= \sum_i e^{-\beta (E_i-\mu N_i)} \end{align} Incidentally, in class we didn't cover the grand potential (or grand free energy), but that is what you get if you try to find a free energy using the Gibbs sum like the partition function.
Fermi-Dirac, Bose-Einstein, and Planck distributions
You should remember these distributions \begin{align} f_{FD}(\varepsilon) &= \frac1{e^{\beta(\varepsilon-\mu)}+1} \\ f_{BE}(\varepsilon) &= \frac1{e^{\beta(\varepsilon-\mu)}-1} \end{align} and should be able to use them to make predictions for properties of non-interacting systems of fermions and bosons. This also requires remembering how to reason about orbitals as essentially independent systems within the grand canonical ensemble. You should remember that the Planck distribution for photons (or phonons) is the same as the Bose-Einstein distribution, but with $\mu=0$. This comes about because photons and phonons are bosons, but are a special kind of boson that has no conservation of particle number.
Density of states
You should remember how to use a density of states together with the above distributions to find properties of a system of noninteracting fermions or bosons \begin{align} \left<X(\varepsilon)\right> &= \int \mathcal{D}(\varepsilon)f(\varepsilon)X(\varepsilon) d\varepsilon \end{align} As special cases of this, you should be able to find $N$ (or given $N$ find $\mu$) or the internal energy. We had a few homeworks where you found entropy from the density of states, but I think that was a bit too challenging/confusing to put on the final exam.
Conditions for coexistence
You should remember that when two phases are in coexistence, their temperatures, pressures, and chemical potentials must be identical, and you should be able to make use of this.

### Equations not to remember

If you need a property of a particular system (the ideal gas, the simple harmonic oscillator), it will be given to you. There is no need, for instance, to remember the Stefan-Boltzmann law or the Planck distribution.

Heat capacity
I do not expect you to remember the definition of heat capacity (although you probably will remember it). \begin{align} C_V &= T\left(\frac{\partial S}{\partial T}\right)_{V,N} \\ &= \left(\frac{\partial U}{\partial T}\right)_{V,N} \\ C_p &= T\left(\frac{\partial S}{\partial T}\right)_{p,N} \end{align} I do expect you to be able to make use of these equations when given. Similarly, you should be able to show that the two expressions for $C_V$ are equal, using the thermodynamic identity.
Enthalpy
If I give you the expression for enthalpy ($U+pV$) you should be able to work with it, but since we didn't touch it in class, I don't expect you to remember what it is.
Any property of an ideal gas
I don't expect you to remember any property of an ideal gas, including its pressure (i.e. ideal gas law), free energy, entropy, internal energy, or chemical potential. You should be comfortable with these expressions, however, and if I provide them should be able to make use of them.
Stefan-Boltzmann equation
You should be able to make use of the expression that \begin{align} I &= \sigma_B T^4 \end{align} where $I$ is the power radiated per area of surface, but need not remember this.
Clausius-Clapeyron equation
You should be able to make use of \begin{align} \frac{dp}{dT} &= \frac{s_g-s_\ell}{v_g-v_\ell} \end{align} but I don't expect you to remember this. You should also be able to convert between this expression and the one involving latent heat using your knowledge of heat and entropy.
Carnot efficiency
You need not remember the Carnot efficiency \begin{align} \epsilon &= 1 - \frac{T_C}{T_H} \end{align} but you should remember what an efficiency is, and should be able to pretty quickly solve for the net work and high-temperature heat for a Carnot engine by looking at it in $T$/$S$ space. (Or similarly for a Carnot refridgerator.)
Density of states for particular systems
You need not remember any expression for the density of states e.g. for a gas. But given such an expression, you should be able to make use of it.
Fermi energy
You need not remember any particular expression for the Fermi energy of a particular system, but should be able to make use of an expression for the Fermi energy of a system.

• face Ideal Gas

face Lecture

120 min.

##### Ideal Gas
Thermal and Statistical Physics 2020

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
• assignment Reduced Mass

assignment Homework

##### Reduced Mass
Central Forces 2023 (3 years)

Using your favorite graphing package, make a plot of the reduced mass $$\mu=\frac{m_1\, m_2}{m_1+m_2}$$ as a function of $m_1$ and $m_2$. What about the shape of this graph tells you something about the physical world that you would like to remember. You should be able to find at least three things. Hint: Think limiting cases.

• 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 Derivatives from Data (NIST)

assignment Homework

##### Derivatives from Data (NIST)
Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
1. Find the partial derivatives $\left(\frac{\partial {S}}{\partial {T}}\right)_{p}$ $\left(\frac{\partial {S}}{\partial {T}}\right)_{V}$ where $p$ is the pressure, $V$ is the volume, $S$ is the entropy, and $T$ is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
2. Why does it take only two variables to define the state?
3. Why are the derivatives above different?
4. What do the words isobaric, isothermal, and isochoric mean?
• assignment Boltzmann probabilities

assignment Homework

##### Boltzmann probabilities
Energy Entropy Boltzmann probabilities Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies $(-\epsilon,0,\epsilon)$.
1. At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy $U$? What is the entropy $S$?
2. At very low temperature, what are the three probabilities?
3. What are the three probabilities at zero temperature? What is the internal energy $U$? What is the entropy $S$?
4. What happens to the probabilities if you allow the temperature to be negative?
• keyboard Electrostatic potential of spherical shell

keyboard Computational Activity

120 min.

##### Electrostatic potential of spherical shell
Computational Physics Lab II 2022

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.
• face Chemical potential and Gibbs distribution

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
• group Charged Sphere

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
• assignment Potential energy of gas in gravitational field

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.
• face Central Forces Introduction: Lecture Notes

face Lecture

5 min.

##### Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)

Learning Outcomes