At what temperature the mean sped of the molecules of hydrogen gas equals the escape speed from the earth?

In kinetic theory of ideal gas mean speed also known as average speed is given as



Where R=gas constant=8.31Jmol-1K-1


T-temperature of gas


M=molar mass of gas


Molar mass of hydrogen=2amu=2g/mol=2kg/mol


Escape speed of earth is the speed given to projectile so that it escapes the gravitational field of earth. It is given by formula



Where G=universal gravitational constant


ME=mass of earth


R=radius of earth


We also know that acceleration due to gravity g is



Multiplying and dividing equation (II) by R



Putting the value of g from equation (III) to above equation we get



According to question vmean is equal to ve


So, from equation (I) and (IV) we get



Radius of earth R=6400km=6400000m


g=9.8m/s



Squaring both sides




Temperature at which the mean sped of the molecules of hydrogen gas equals the escape speed from the earth is 11800K.


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