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.

      Plot of magnetization vs. B field

      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 Mass of a Slab

      assignment Homework

      Mass of a Slab
      Static Fields 2022 (5 years)

      Determine the total mass of each of the slabs below.

      1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
      2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
      3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
      4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
      5. What are the dimensions of \(A\)?
      6. Write several sentences comparing your answers to the different cases above.

  • 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}

    Effects of High Altitude by Randall Munroe, at xkcd.