# Ideal gas temperature and pressure gradients?

Consider an ideal gas in a $d\times d\times L$ box with the $L$ dimension in the $x$-direction. Suppose that the opposite $d\times d$ sides of the box are held at temperatures $T_1$ and $T_2$ with $T_2>T_2$ and that the system reaches a steady state. According to these notes, the thermal conductivity of an idea gas scales as the square root of temperature; $k=\alpha\sqrt{T}$ in which case by Fourier's Law one gets that the temperature gradient in the $x$-direction is $$T(x) = \left[T_1^{3/2}+(T_2^{3/2}-T_1^{3/2})\frac{x}{L}\right]^{2/3}$$ What is the corresponding pressure gradient $P(x)$ in the steady state?

-