So I'm trying to calculate the concentration of an ideal gas in an adiabatic container as a function of position where the top and bottom plates of the container are fixed at temperatures T1 and T2, respectively, with T2 > T1. Thus, there is a temperature gradient from bottom to top, ie. $T(z) = T_2 - (T_2-T_1)\frac{z}{h}$, where h is the height of the container. How do I calculate the concentration of the gas, $n(z)$, which I expect to be smaller at the bottom (near the hot plate) and larger at the top?

EDIT: I started with a similar problem but which included the gravitational field. The correct insight was to realize that the difference in pressure between z and z+dz was equal to the gravitational force due to the mass in the volume between z and z+dz. From this it's easy to follow the math. In the present case, however, is the correct insight simply that the pressure is uniform? ie. since there is no mass to balance, there are no pressure differences? If so, I have found the solution: $n(z) = n(0)T_2(T_2 - (T_2-T_1)\frac{z}{h})^{-1}$, which is inversely proportional to the temperature, and which seems kind of nice, but how can I properly justify that the pressure is uniform?

I'm trying to calculate the concentration of an ideal gas in an adiabatic container as a function of position where the top and bottom plates of the container are fixed at temperatures $T_1$ and $T_2$, respectively, with $T_2 > T_1$. Thus, there is a temperature gradient from bottom to top, i.e. $T(z) = T_2 - (T_2-T_1)\frac{z}{h}$, where $h$ is the height of the container. How do I calculate the concentration of the gas, $n(z)$, which I expect to be smaller at the bottom (near the hot plate) and larger at the top?

EDIT: I started with a similar problem but which included the gravitational field. The correct insight was to realize that the difference in pressure between $z$ and $z+dz$ was equal to the gravitational force due to the mass in the volume between $z$ and $z+dz$. From this it's easy to follow the math. In the present case, however, is the correct insight simply that the pressure is uniform? I.e., since there is no mass to balance, there are no pressure differences? If so, I have found the solution: $n(z) = n(0)T_2(T_2 - (T_2-T_1)\frac{z}{h})^{-1}$, which is inversely proportional to the temperature, and which seems kind of nice, but how can I properly justify that the pressure is uniform?

So I'm trying to calculate the concentration of an ideal gas in an adiabatic container as a function of position where the top and bottom plates of the container are fixed at temperatures T1 and T2, respectively, with T2 > T1. Thus, there is a temperature gradient from bottom to top, ie. $T(z) = T_2 - (T_2-T_1)\frac{z}{h}$, where h is the height of the container. How do I calculate the concentration of the gas, $n(z)$, which I expect to be smaller at the bottom (near the hot plate) and larger at the top?

So I'm trying to calculate the concentration of an ideal gas in an adiabatic container as a function of position where the top and bottom plates of the container are fixed at temperatures T1 and T2, respectively, with T2 > T1. Thus, there is a temperature gradient from bottom to top, ie. $T(z) = T_2 - (T_2-T_1)\frac{z}{h}$, where h is the height of the container. How do I calculate the concentration of the gas, $n(z)$, which I expect to be smaller at the bottom (near the hot plate) and larger at the top?

EDIT: I started with a similar problem but which included the gravitational field. The correct insight was to realize that the difference in pressure between z and z+dz was equal to the gravitational force due to the mass in the volume between z and z+dz. From this it's easy to follow the math. In the present case, however, is the correct insight simply that the pressure is uniform? ie. since there is no mass to balance, there are no pressure differences? If so, I have found the solution: $n(z) = n(0)T_2(T_2 - (T_2-T_1)\frac{z}{h})^{-1}$, which is inversely proportional to the temperature, and which seems kind of nice, but how can I properly justify that the pressure is uniform?