Entropy, energy, and enthalpy of van der Waals gas

• Van der Waals gas Enthalpy Entropy
• assignment Vapor pressure equation

  assignment Homework

  Vapor pressure equation

  phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.

  1. Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.

  2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

• face Phase transformations

  face Lecture

  120 min.

  Phase transformations

  Thermal and Statistical Physics 2020

  phase transformation Clausius-Clapeyron mean field theory thermodynamics

  These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
• assignment Magnetic susceptibility

  assignment Homework

  Magnetic susceptibility

  Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020

  Consider 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.

  

  1. 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 \(x\equiv \frac{mB}{kT}\).

  2. 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.

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

• assignment Ideal gas calculations

  assignment Homework

  Ideal gas calculations

  Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

  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.

  1. How much heat (in joules) is added to the gas in each of these two processes?

  2. What is the temperature at the end of the second process?

  3. 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?

• face Boltzmann probabilities and Helmholtz

  face Lecture

  120 min.

  Boltzmann probabilities and Helmholtz

  Thermal and Statistical Physics 2020

  ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

  These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
• face Work, Heat, and cycles

  face Lecture

  120 min.

  Work, Heat, and cycles

  Thermal and Statistical Physics 2020

  work heat engines Carnot thermodynamics entropy

  These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
• face Ideal Gas

  face Lecture

  120 min.

  Ideal Gas

  Thermal and Statistical Physics 2020

  ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

  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 Calculation of $\frac{dT}{dp}$ for water

  assignment Homework

  Calculation of \(\frac{dT}{dp}\) for water

  Clausius-Clapeyron Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of \(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor equilibrium of water. The heat of vaporization at \(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result in kelvin/atm.
• 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.
• assignment Ideal gas in two dimensions

  assignment Homework

  Ideal gas in two dimensions

  Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020

  1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

  2. Find an expression for the energy \(U\) of the gas.

  3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

• Thermal and Statistical Physics 2020

  In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.

  1. 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}

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

  3. 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}