What is the molecular mass of the unknown gas in the atmosphere?

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To calculate the molecular mass of an unknown gas in the atmosphere, the user is working with temperature, gravitational constants, and pressure values at different altitudes. They have derived the gravitational acceleration using the formula g = GM/R^2, but are struggling to connect this with the molecular mass equation g = kT/mλ. The calculated gravitational value is significantly lower than Earth's, suggesting a smaller mass for the unknown gas. Despite rearranging the equations, the resulting molecular mass is off by a factor of 10^3 compared to known gases. Further guidance on integrating gravitational constants with molecular mass calculations is needed.
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Homework Statement



I need to calculate the mas of a molecule of an unknown gas in the atmosphere from the values:

T = 500 K
mass of the planet = 1.5 × 1024 kg
radius is 3.0 × 106 m
universal constant of gravitation, G = 6.67 × 10^−11 N m^2 kg^−2
P at ground level = 2.0 × 10^7 Pa
P at 50km altitude = 2.0 × 10^5 Pa

Homework Equations



g=kT/mλ

but I can't find an equation involving G :(

The Attempt at a Solution



I am completely stuck, if I can get ahint where to start I'll have a go and see if I get anywhere.
 
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On Earth

F = mg = GmM/R^2 so g = GM/R^2
 
Right, I've got so far as calculating g now and have calculated a very low value for g compared to Earth which is consistent with having a smaller mass than earth. I've also worked out an equation for m by rearranging λ=kT/mg ut the value doesn't match that of the two gases it could be by a factor of 10^3.
 
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