(i) Show that the equation of state of an ideal gas is still PV = RT even when the gas is heated to such a high temperature that the particles are moving at relativistic speeds. (Hint: What feature of the partition function of the ideal gas determines the gas law?).
(ii) Although the equation of state does not alter when the particles in a monatomic ideal gas start to move at relativistic speeds, show that in the formula for an adiabat, PV = constant, lambda in the relativistic limit is 4/3, rather than 5/3 as in the non-relativistic case.