What is the height of the methane atmosphere?

[ghostbuster25]

sounds simple, I am however stuck

A planet has a thin isothermal atmosphere at a temperature of 320K and is composed entirely of methane (relative molecular mass of 16)
The acceleration due to gravity at the surface is 8.3 m/s/s
calculate the height of the atmosphere.

not really sure where to start with this one

 

[DeShark]

What is the escape velocity?
What is the velocity of the methane molecules?

 

[RoyalCat]

What is the escape velocity?
What is the velocity of the methane molecules?

The two approaches may be equivalent, but I think that what the question is actually about is thermodynamics and hydrostatics rather than gravitation.
You need radius/mass (Assuming $$G$$ is known) of the planet to calculate the escape velocity. Even then it wouldn't be that good of a criterion seeing how the velocity you'd find based on the temperature is just the average velocity. Though the final solution to the isothermal atmosphere leaves a bit of ambiguity as to where the atmosphere 'ends' just as well...

You are told that the atmosphere is thin, therefore the difference in gravity between the surface and the edge of the atmosphere is negligible in this problem.
Construct a model of an isothermal atmosphere assuming that the pressure in the atmosphere is hydrostatic.

The formula you might not remember is $$\frac{dP}{dh}=-\rho g$$ where $$\rho$$ and $$g$$ both depend on the height (In our case, $$g$$ is a constant, an interesting exercise would be to let it vary with height according to Newton's Law of Gravitation, but that would require knowing the radius of the planet)