B.1 An ideal gas of N ultrarelativistic particles is confined to a volume V. The gas is in thermal equilibrium with a heat bath at temperature T. The energy of each particle with linear momentum p is given by ?(p) = cp, where p = |p| and c is the speed of light in vacuum. a) Derive the Maxwell-Boltzmann distribution for the probability of a single particle to have its momentum between p and p + dp. What is the most probable value of p? b) Calculate the single-particle partition function Z(1). c) Employ the semi-classical approximation to show that the partition function of the gas is given by Z = V^N / (?^{2N} N!) * (k_B T / ?c)^{3N}, where ? and k_B are the reduced Planck and Boltzmann constants, respectively. Explain the origin of the 1/N! factor.
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In this case, the energy of each particle is given by e(p) = cp, so we can substitute this into the Boltzmann factor: f(p) = e^(-cp/kT) To find the probability of a single particle to have its momentum between p and p + dp, we need to integrate the Boltzmann Show more…
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Sri K.
According to Maxwell-Boltzmann statistical count, the total number of arrangements of the particles is given by: W = N! ̧̣̀̑̐̓_{r=1}^s [g_r^{n_r} / n_r!] a) Derive the general form of Maxwell Boltzmann energy distribution law. b) By using the continuous variation of free particles in an ideal gas, show that the expression of the multiplier ̲ is: ̲ = log [V/N (2̰mkT / h^2)^(3/2)] Given that, e^-̲ = N / ∑ g_r e^(-E_r / kT) c) Deduce the Maxwell Boltzmann energy distribution law for particles in an ideal gas. d) Show that the entropy: S = kN log Z + (U/T), where Z is the partition function Z = ∑ g_r e^-̱ E_r. e) Find the most probable and the root mean square speed of N₂ molecule at 27
Consider the air in an oven at 500 K. The oven has a volume of 0.15 m^3 and contains 2.2 × 10^24 identical nitrogen molecules, each having five degrees of freedom and a mass of 4.8 × 10^−26 kg. (a) What is the thermal energy of this system? (b) The magnitude of the momentum of any molecule can range from 0 to p0. Estimating p0 to be roughly twice the root mean square momentum, what is the volume in momentum space that is available to any particle? (c) We are going to calculate the number of accessible quantum states for the molecules' translational motions, ignoring the rotational states because the latter turn out to be relatively few in comparison. Considering the translational motion only, how many different quantum states would be accessible to any particle, if it were all by itself? (VrVp/h^3, where Vp = (4/3)πp^3/0.) (d) What is the number of states per particle ωc corrected for the case of identical particles? (e) How many different quantum states are accessible to the entire system?
Suman K.
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