V From the Sackur-Tetrode formula S(E, V, N) = Nk ln 3/27 +-Nk, derive Απm E %3D 3H3 N s many properties of the monotonic ideal classical gas as you can.
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- Imagine a photon gas at an initial temperature of T = 1.4 K. What is the temperature of the photon gas (in K) after it has undergone a reversible adiabatic expansion to 2 times its original volume?Consider a system with 1000 particles that can only have two energies, ɛ, and with ɛ, > E,. The difference between these two values is Aɛ = ɛ, -& . Assume that gi = g2 = 1. Using the %3D %3D equation for the Boltzmann distribution graph the number of particles, ni and m, in states & n2, E and E, as a function of temperature for a Aɛ = 1×10-2' J and for a temperature range from 2 to 300 K. (Note: kg = 1.380x10-23 J K-!. %3D %3D (s,-s,) gLe Aɛ/ n2 or = e n,Consider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac- teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from all other internal modes. Calculate the canonical partition function, the average energy, and the heat capacity at constant volume, Cv.
- Show that the one-particle partition function Z₁ for a 2D ideal gas confined to area A is: A 2²/1 Z₁ = SThe exact differential for the Gibbs energy is given by dG = -SdT + VdP. The form of this differential implies which of the following relationships? O (7),- (), ƏG Әт ƏG др P T (37), - - (-), P T as ( x) - (*), = др T (327), = -(0)₁ == P TA diatomic gas molecule can be in one of two vibrational energy levels, separated by 0.1 eV. Give the probabilities to be in either state and use these to calculate their relative populations at room temperature, T≈ 300 K. [You may use that kB ≈ 8.6 × 10−5 K eV−1]