Determining temperature of an ideal gas

Question

Response to the following problem:

The temperature of an ideal gas in a tube of very small, constant cross-sectional area varies linearly from one end (x = 0) to the other end (x = L) according to the equation

T = (T₀ + T_L - T₀/L)x

If the volume of the tube is V and the pressure P is uniform throughout the tube, show that the equation of state for n moles of gas is given by

PV = nR (T_L - T₀/In(T_L/T₀))

Show that, when T_L = T_o = T, the equation of state reduces to the obvious one, P_V = nRT.