Question

The temperature at which the root mean square speed is equal to the speed of escape from the surface of the earth for

• A. nitrogen is $1.409\times {10}^{5}K$
• B. nitrogen is $7.045\times {10}^{4}K$
• C. hydrogen is 5032 K
• D. hydrogen is 10063 K

Answer 1

Answer: (B), (C)

The correct options are
B nitrogen is $7.045\times {10}^{4}K$
C hydrogen is 5032 K

The escape velocity from surface of the earth $V_e=\sqrt{2gR}g=10m/s^{2}R=6.4\ast {10}^{6}m$

Root mean square speed$u_{rms}=\sqrt{\frac{3RT}{M}}$

According to given condition,

$\frac{3RT}{M}=2g{R}_e\Rightarrow T=\frac{2g{R}_{e}M}{3R}$

For nitrogen,

$T=\frac{2g{R}_{e}M}{3R}=7.045\times {10}^{4}K$

For Hydrogen,

$T=\frac{2g{R}_{e}M}{3R}=5032K$

Hence option B& C is correct