- Energy Temperature Paramagnetism
*face*Entropy and Temperature*face*Lecture120 min.

##### Entropy and Temperature

Thermal and Statistical Physics 2020paramagnet entropy temperature statistical mechanics

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.*assignment*Nucleus in a Magnetic Field*assignment*Homework##### Nucleus in a Magnetic Field

Energy and Entropy 2021 (2 years)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.

*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*Entropy of mixing*assignment*Homework##### Entropy of mixing

Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

Show that after diffusive equilibrium is reached the total entropy is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is known as the entropy of mixing.

If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

Since the Helmholtz free energy is lower for the mixed \(AB\) than for the separated \(A\) and \(B\), it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

- Note
This course has not yet covered

*work*, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

*face*Chemical potential and Gibbs distribution*face*Lecture120 min.

##### Chemical potential and Gibbs distribution

Thermal and Statistical Physics 2020chemical 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.*assignment*Series Notation 1*assignment*Homework##### Series Notation 1

Static Fields 2023 (6 years)Write out the first four nonzero terms in the series:

\[\sum\limits_{n=0}^\infty \frac{1}{n!}\]

- \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
- \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

*assignment*Entropy, energy, and enthalpy of van der Waals gas*assignment*Homework##### Entropy, energy, and enthalpy of van der Waals gas

Van der Waals gas Enthalpy Entropy Thermal and Statistical Physics 2020In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.

Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

*assignment_ind*Magnetic Moment \& Stern-Gerlach Experiments*assignment_ind*Small White Board Question30 min.

##### Magnetic Moment & Stern-Gerlach Experiments

Quantum Fundamentals 2023 (3 years)Angular Momentum Spin Magnetic Moment Stern-Gerlach Experiments

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.*assignment*Magnetic susceptibility*assignment*Homework##### Magnetic susceptibility

Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020Consider a paramagnet, which is a material with \(n\) spins per unit volume each of which may each be either “up” or “down”. The spins have energy \(\pm mB\) where \(m\) is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization \(M\) is defined as the total magnetic moment divided by the total volume.

*Hint:*each individual spin may be treated as a two-state system, which you have already worked with above.Find the Helmholtz free energy of a paramagnetic system (assume \(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of only the ratio \(x\equiv \frac{mB}{kT}\).

Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization \(M\) (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.

Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the limit \(mB\ll kT\).

*face*Basics of heat engines*face*Lecture10 min.

##### Basics of heat engines

Contemporary Challenges 2021 (4 years) This brief lecture covers the basics of heat engines.-
Thermal and Statistical Physics 2020
Find the equilibrium value at temperature \(T\)
of the fractional magnetization \begin{equation}
\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}
\end{equation} of a system of \(N\) spins each of magnetic moment
\(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy
of this system is given by \begin{align}
U &= -\mu_{tot}B
\end{align} where \(\mu_{tot}\) is the total magnetization. Take the
entropy as the logarithm of the multiplicity \(g(N,s)\) as given in
(1.35 in the text): \begin{equation}
S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}
\end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which
is related to the magnetization by \(\mu_{tot} = 2sm\).
*Hint*: Show that in this approximation \begin{equation} S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N}, \end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that \(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes \(\langle U\rangle\), the thermal average energy.