Vapor pressure equation

  • phase transformation Clausius-Clapeyron
    • 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.
    • 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 Heat of vaporization of ice

      assignment Homework

      Heat of vaporization of ice
      Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor pressure of water at its triple point is 611 Pa, at 0.01\(^\circ\text{C}\) (see Estimate in \(\text{J mol}^{-1}\) the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?
    • face Review of Thermal Physics

      face Lecture

      30 min.

      Review of Thermal Physics
      Thermal and Statistical Physics 2020

      thermodynamics statistical mechanics

      These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
    • assignment Basic Calculus: Practice Exercises

      assignment Homework

      Basic Calculus: Practice Exercises
      Static Fields 2023 (4 years) Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
      1. \(\frac{d}{du}\left(u^2\sin u\right)\)
      2. \(\frac{d}{dz}\left(\ln(z^2+1)\right)\)
      3. \(\displaystyle\int v\cos(v^2)\,dv\)
      4. \(\displaystyle\int v\cos v\,dv\)
    • 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 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.

    • face Energy and heat and entropy

      face Lecture

      30 min.

      Energy and heat and entropy
      Energy and Entropy 2021 (2 years)

      latent heat heat capacity internal energy entropy

      This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
    • assignment Adiabatic Compression

      assignment Homework

      Adiabatic Compression
      ideal gas internal energy engine Energy and Entropy 2020

      A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

      In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

      Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

      1. If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

      2. By what factor does the pressure increase?

    • 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?

  • 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).