# Non uniform temperature in a gas

Michael Owen
Hi all,

Supposed if I have an air tight container, containing air with an initial pressure p0 and temperature T0. If I heat up the container from its bottom, and assuming its volume does not change, the temperature in the constainer will become non-uniform (highest at the bottom and lowest at the top). So how about the pressure in the container? Does it have uniform pressure or non-uniform (highest at the bottom and lowest at the top)? Can I do the calculation using ideal gas law (P/T=constant)?

Thanks,
Owen

## Answers and Replies

Oudeis Eimi
Hi all,

Supposed if I have an air tight container, containing air with an initial pressure p0 and temperature T0. If I heat up the container from its bottom, and assuming its volume does not change, the temperature in the constainer will become non-uniform (highest at the bottom and lowest at the top). So how about the pressure in the container? Does it have uniform pressure or non-uniform (highest at the bottom and lowest at the top)? Can I do the calculation using ideal gas law (P/T=constant)?

Thanks,
Owen

This is an interesting question. First, assuming the container is at rest WRT an intertial
frame, but subject to gravity (as would be approximately the case for, say, a cup
resting on your kitchen table), the pressure in the fluid (and the surrounding air) will
be non uniform due to gravity. Thus, you are forced from the onset to work with
a pressure field. Then, for most practical purposes, and except for very rapid changing
conditions (both in time and space), you can consider a situation of local thermodynamic
equilibrium, that is, promote the thermodynamic variables to fields and assume that
thermo stays valid locally ("pointwise") for the fields.

For instance, for an ideal gas, the equation of state would be p = rho kB T, where
p = p(x, y, z, t) pressure, rho = rho(x, y, z, t) density and T = T(x, y, z, t) are
macroscopic fields.

--- EDIT ---

As the poster after me said, the setup described above will lead to air flow due to
convection. In general, the absence of global equilibrium inside the vessel will
be accompanied by fluxes - heat and mass flux in the example under discussion.
The problem is then transformed into one of fluid mechanics, involving heat
transport by both conduction and convection.

Last edited:
poso
Heating the container from the bottom will drive a flow of gas inside the container. You might call this a flow occurring due to natural convection. There is only flow possible if there are pressure gradients inside your container. If the driving (bottom) temperature is very high and the top of the container is kept very cold you might expect large temperature gradients inside the gas and corresponding large gas flow across the container volume. Please remember that the temperature in the ideal gas law is expressed in Kelvin so e.g. 20 C = 294 K and 40 C = 313 K You can now easily see that the average temperature expressed in Kelvin 303.5 K does not deviate that much from 294 K and 313K (about 3%). Small enough to consider the gas as ideal.