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- 1: Speed of sound in a fluid. When a sound wave passes through a fluid the period of vibration is short compared to the relaxation time for a small volume of the fluid to exchange heat with the rest of the fluid and the compression of such an element of volume can be considered adiabatic. Show that in 1D compressions and rarefactions of the fluid contained in a slab of thickness dx the pressure p(x, t) in the fluid satisfies the wave equation
∂2p/∂t 2 = u2 ∂2p/∂x2 ,
where the velocity of sound propagation u is a constant given by \(u = (\rho_0^~\kappa_S^~)^{-1/2}\), \(\rho_0^~\) is the equilibrium mass density of the fluid, and \(\kappa_S^~\) its adiabatic compressibility. [Hint: The slab of thickness dx feels a pressure p(x) on one side and p(x+dx) on the other side, and contains particles with mass m, number density n(x) and mean velocity v(x); Write down Newton's second law for the slab, and the continuity equation for the mass density and mass current density; Combine these two (by taking further derivatives) into one equation for p(x).]
- 2: Equipartition Principle [Kennett, Problem 4.7]
Use the equipartition principle to estimate the typical angular velocity of a nitrogen molecule at room temperature. (The N-N bond length is 1.0975 Å and the mass of an N atom is 14.0067 amu.)
- 3: Relativistic ideal gas in a canonical state
Consider an extreme relativistic gas consisting of N monatomic
particles with energy-momentum relationship ε = pc, c being the speed of light. (a) Show that the partition function ZN is given by
ZN(V, T) = (1/N!) {8πV (kBT/hc)3}N.
(b) Find the energy and the pressure equations of state for this system.
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