Question

The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth is closest to
(Take $g=10m/s^2$)

• A. ${10}^4\text{K}$
• B. $650\text{K}$
• C. $3\times {10}^5\text{K}$
• D. $800\text{K}$

Answer 1

The correct option is A ${10}^4\text{K}$

We know that,
Escape velocity, $V_e=\sqrt{2gR}[R\to \text{radius of earth}]\therefore V_e=\sqrt{2\times 10\times 6400\times {10}^3}[R=6400\text{km}=6400\times {10}^3\text{m}]V_e\approx 11314\text{m/s}$
Also, $V_{rms}=\sqrt{\frac{3RT}{M}}\Rightarrow 11314=\sqrt{\frac{3\times 8.3\times T}{2\times {10}^{-3}}}$
[Given $V_e=V_{rms}$, Molar mass of hydrogen, $M=2\text{g}$]
$\Rightarrow T=\frac{{11314}^2\times 2\times {10}^{-3}}{3\times 8.3}\Rightarrow T=10281\text{K}\approx {10}^4\text{K}$