# Temperature and pressure gradient in a gas

I have derived that, when there is a temperature difference (gradient) in a gas (consider a long tube with one end maintained at 100oC and other end maintained at 0oC), there will be a pressure gradient (something similar to Bernoulli's law).
I would like to know the feedback from the community on this.

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The analysis is incorrect. There will be a linear temperature gradient from A to B but the pressure will be constant. If the pressure is constant, the molar density at point x along the tube will be ##\frac{P}{RT(x)}##. So the total number of moles in the tube will be $$n=\int_0^L{\frac{P}{RT(x)}Adx}=\frac{PV}{R}\frac{1}{L}\int_0^L{\frac{dx}{T(x)}}$$ where A is the cross sectional area of the tube and V is the tube volume.