Review Week
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 \(UTS\), 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.
 FermiDirac, BoseEinstein, 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 noninteracting 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 BoseEinstein 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 StefanBoltzmann 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.
 StefanBoltzmann 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.
 ClausiusClapeyron equation

You should be able to make use of \begin{align}
\frac{dp}{dT} &= \frac{s_gs_\ell}{v_gv_\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 hightemperature
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.