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Find the minimum radius of planet with mean density $\rho$ and temperature $T$ which can detain Oxygen in its atmosphere and $G$ = Universal Gravitational Constant and $M$ is molecular mass of Oxygen gas.

I tried to do it with the kinetic energy of oxygen molecule as $K.E$=$\frac{5mRT}{2M}$ and then equating it with the potential as $\frac{GMm}{R_{planet}}$ but I am getting the wrong answer. Here I assumed that mass of gas is $m$. $$\frac{5mRT}{2M}=\frac{G\rho \frac{4\pi}{3}R^3m}{R_{planet}}$$ Whats wrong with this method?

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You didn't use the density provided to calculate the planetary mass based on radius. I think you need to consider the effect of gravity a single molecule of mass M (from the question) so the energy can equate to the KE calculation. the other m should be the mass of the planet.

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  • $\begingroup$ I used the mean density While writing the mass of planet as $rho$=$\frac{Mass of planet}{Volume of planet}$ and why would need to consider effect of gravity on single particle because even if we consider the mass m it would get cancelled from both sides.?? $\endgroup$ – Piyush Raut Jan 17 '17 at 13:49
  • $\begingroup$ try KE$=3kT/2$ for one molecule of mass M $\endgroup$ – JMLCarter Jan 17 '17 at 17:38
  • $\begingroup$ yes that would give the answer. buy why to consider only translational degrees of freedom? $\endgroup$ – Piyush Raut Jan 17 '17 at 18:15
  • $\begingroup$ The concept you want to understand is that only translation of the whole molecule can act against gravity. $\endgroup$ – JMLCarter Jan 17 '17 at 22:20

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