Question

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

Use R = 8.314 JK^-1 mol^-1

Answer 1

Mean speed of the molecule is givne by

\[\sqrt{\frac{8RT}{\pi M}}\] 

\[\text { For  H  molecule },   M   = 2 \times  {10}^{- 3} kg  \] 

\[ = \sqrt{\frac{4RT \times {10}^3}{\pi}}\]

For escape velocity of Earth :-

Let r be the radius of Earth

\[v = \sqrt{\frac{2GM}{r}}\] 

Multiplying numerator and denominator by R, we get

\[ v_c  = \sqrt{\frac{GM}{r^2}2r}\] 

\[g = \frac{GM}{r^2}\] 

\[ v_c  = \sqrt{2gr}\]

\[\sqrt{\frac{4RT \times {10}^3}{\pi}} = \sqrt{2gr}\] 

\[ \Rightarrow \frac{2 \times 8 . 314 \times T \times {10}^3}{3 . 142} = 9 . 8 \times 6 . 37 \times  {10}^6 \] 

\[ \Rightarrow T \approx 11800  \text { K }\]