Find the minimum radius of the planet of density `5.5 times 10^3 kg//m^3` and temperature `427^@C` which can hold `O_2` in its atmosphere. [Given`G=6.67times 10^-11 N.m^-2.kg^-2` and `R=8.3J.mol^-1.K^-1`]

Explanation

Escape velocity of any object at the surface of a planet of radius r and mass `M_1` is
`v=sqrt((2GM_1)/r)=sqrt((2G)/r.4/3pir^3p)`
[where p is the material density of the planet]
`=sqrt(8/2Gpir^2p)`
rms speed of a gas of molecular weight M at an absolute temperature T is c `=sqrt((3RT)/M)`
SInce, the planet holds `O_2`, hence `v_(min)=c`
`thereforesqrt(8/3Gpir^2minrho)=sqrt((3RT)/M)`
[`r_(min)`-minimum radius of the planet]
or, `8/3Gpir^2minp=(3RT)/M`
or,`r_(min)^2=(9RT)/(8GpipM)`
or,`r_(min)=sqrt((9RT)/(G pipM))`
`=sqrt((9times9.3times(427+273)times7)/(8times6.67times10^-11times22times5.5times10^3times32times10^-3))`
`=421 times 10^3 m=421 km`