face Lecture
120 min.
Gibbs entropy information theory probability statistical mechanics
These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.face Lecture
120 min.
chemical potential Gibbs distribution grand canonical ensemble statistical mechanics
These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.face Lecture
30 min.
thermodynamics statistical mechanics
These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.assignment Homework
assignment Homework
Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).
Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).
group Small Group Activity
30 min.
assignment Homework
Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]
Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].
assignment Homework
assignment Homework
Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.
How much heat (in joules) is added to the gas in each of these two processes?
What is the temperature at the end of the second process?
Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?
assignment Homework
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 \(\varepsilon\).
From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.
Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.
Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).
Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.
Solve for the thermal average occupancy of the system in terms of \(\lambda\).
Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}
Find an expression for the thermal average energy of the system.
Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.