# Conceptual Problem in Kinetic Theory of Gas

I came across a question from I.E. Irodov's Physics book: A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free fall acceleration is equal to g. The temperature of the nitrogen varies along the height h so that its density is the same throughout the volume. What is the temperature gradient dT/dh? The solution was this:

Now, my question is, how can we use the ideal gas equation and the equation for pressure variation in a liquid with depth simultaneously? The ideal gas equation is valid only for those gases that can be modelled with the kinetic theory of gases, so how can we apply the other equation on this gas?

• The top part talks about nitrogen in a gaseous state. I don't understand bringing up the liquid, but, if so, at some height there would be an interface between liquid and gas, you would use the proper equations for each Oct 12 '20 at 2:17
• You’re missing a minus sign. Oct 12 '20 at 2:18

The equation $$P= \rho gh$$ is for ideal fluids not just for liquids. The postulates for the pressure equation are far lesser than the postulates in the ideal gas equation. However, since the two sets of postulates don't contradict each other, we can apply the pressure equation to an ideal gas, even though we cant apply the ideal gas equation to an ideal liquid. Because an ideal gas already fulfills the conditions for the pressure equation, while also fulfilling further conditions for the gas equation.