Answer Happy

In the lecture notes (part 1) I displayed the equation for the pressure of a quantum ideal gas in the form 2 4π (ε₁ / c²-m²c²) ³¹² S 3h³ m₁c² - P₁ = g₁ ARV | p²c² ƒ (5₂) p² dp = 8₁, 8A h³ 3V & p 0 ● 4π Use the following arguments to derive both parts of this equation: Use the thermodynamic relation dE=-PdV+TdS to derive an equation for pressure (as a partial derivative). use the previous equation in the notes for energy density of the gas 00 h³ -HA/KT ±1 Ep-HA 0 -dep EA = 8A -§ε‚ƒ (ε₂) p²dp and assuming that the number of particles in each quantum state remains fixed (assuming that the distribution function does not change with volume) and that internal energy of the gas changes only because the energy of each quantum state & depends on the volume V, write the pressure as an integral. Since the wave vector k x L x V-1/³, find the dependence of particle momentum p on volume and then find the dp/dV, and next using the relativistic equation of energy 2 ²p² + m²c² Ep =+√√√ find the derivative dɛ/dp. Now find de/dV using the chain rule and substitute into the integral in part (b) to arrive at the first equation for pressure in the notes. Next use the relativistic equation for particle energy to arrive at the second equation