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

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SUMMARY

The discussion focuses on calculating the molecular mass of an unknown gas in the atmosphere using the values of temperature (T = 500 K), gravitational constant (G = 6.67 × 10^−11 N m^2 kg^−2), and pressure at different altitudes. The participant successfully derived the gravitational acceleration (g) using the formula g = GM/R^2 but encountered discrepancies in the calculated molecular mass when applying the equation λ = kT/mg. The participant seeks guidance on how to incorporate the universal gravitational constant into their calculations effectively.

PREREQUISITES
  • Understanding of the ideal gas law and molecular mass calculations
  • Familiarity with gravitational physics and the formula F = mg
  • Knowledge of pressure variations with altitude
  • Basic proficiency in algebraic manipulation of equations
NEXT STEPS
  • Research the relationship between pressure, temperature, and molecular mass using the ideal gas law
  • Study the derivation and application of the equation g = GM/R^2 in atmospheric science
  • Explore the concept of mean free path (λ) in relation to gas behavior
  • Investigate the effects of altitude on pressure and density in the atmosphere
USEFUL FOR

Students studying atmospheric physics, researchers in gas dynamics, and anyone interested in calculating molecular masses of gases in varying atmospheric conditions.

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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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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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