Pressure of thermal radiation

• Thermal radiation Pressure
• face Thermal radiation and Planck distribution

  face Lecture

  120 min.

  Thermal radiation and Planck distribution

  Thermal and Statistical Physics 2020

  Planck distribution blackbody radiation photon statistical mechanics

  These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• assignment Gibbs entropy is extensive

  assignment Homework

  Gibbs entropy is extensive

  Gibbs entropy Probability Thermal and Statistical Physics 2020

  Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

  1. Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
  2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

  Note

  In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

• face Energy and Entropy review

  face Lecture

  5 min.

  Energy and Entropy review

  Thermal and Statistical Physics 2020 (3 years)

  thermodynamics statistical mechanics

  This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
• assignment Derivatives from Data (NIST)

  assignment Homework

  Derivatives from Data (NIST)

  Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.

  1. Find the partial derivatives \[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\] \[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\] where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
  2. Why does it take only two variables to define the state?
  3. Why are the derivatives above different?
  4. What do the words isobaric, isothermal, and isochoric mean?
• 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?
• group Heat and Temperature of Water Vapor

  group Small Group Activity

  30 min.

  Heat and Temperature of Water Vapor

  Thermo Heat Capacity Partial Derivatives

  In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• 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).

• assignment Isothermal/Adiabatic Compressibility

  assignment Homework

  Isothermal/Adiabatic Compressibility

  Energy and Entropy 2021 (2 years)

  The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

• group Using $pV$ and $TS$ Plots

  group Small Group Activity

  30 min.

  Using \(pV\) and \(TS\) Plots

  Energy and Entropy 2021 (2 years)

  work heat first law energy

  Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
• 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.
• Thermal and Statistical Physics 2020

  (modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

  1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where \(\langle n_j\rangle\) is the number of photons in the mode \(j\);

  2. Solve for the relationship between pressure and internal energy.