Question

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

Answer 1

Hint: The mean speed of the molecules is the average velocity shown by the molecules of gas in a unit square area. It is dependent on both the mass and temperature of the gas. The escape speed is the minimum velocity with which the body should be projected from the Earth's surface so that it escapes from the Earth's gravitational field.

Complete answer:
The formula for the mean speed of the gas is given by,
$\Rightarrow Mean\text{ speed of gas molecule = }\sqrt{\dfrac{8RT}{\pi M}}$
Where,
R is the gas constant
T is the temperature in Kelvin
M is the molar mass of gas
Also for Escape velocity,
Let r be the radius.
$\Rightarrow Velocity\text{ }=\sqrt{\dfrac{2GM}{r}}$
Multiply the numerator and denominator by ‘r’, we get,
${{v}_{e}}\text{ = }\sqrt{\dfrac{2GMr}{r\times r}}$
But we know that,$g\text{ = }\dfrac{GM}{{{r}^{2}}}$
Substituting the formula for ‘g’ in the formula for escape velocity,
${{v}_{c}}\text{ = }\sqrt{2gr}$
We have been given that the escape velocity of the Earth is equal to the mean speed of hydrogen gas.
Gas constant=R= 8.314 J/K mol
Radius of Earth= r= $64\times {{10}^{-5}}$km
Mass of Hydrogen= M= $2\times {{10}^{-3}}$
Acceleration due to gravity= g= 9.8 m/s2
Mean speed of molecule = Escape velocity
$\sqrt{\dfrac{8RT}{\pi M}}\text{ = }\sqrt{2gr}$
Squaring both the sides we get,
$\text{ }\dfrac{8RT}{\pi M}=\text{ }2gr$
$T\text{ = }\dfrac{2gr\pi M}{8R}$
Substituting the values in the above formula,
$T\text{ = }\dfrac{2\times 9.8\times 64\times {{10}^{5}}\times 3.14\times 2\times {{10}^{-3}}}{8\times 8.3}$
$\begin{align}
  & T\text{ = 11863}\text{.9 K } \\
 & \Rightarrow T\approx \text{11800 K} \\
\end{align}$
The mean speed of the molecules of the hydrogen gas equals the escape speed from the Earth at 11800 K.

Note:
The speed of the gas molecules is directly proportional to the temperature (K) and inversely proportional to the molar mass of the gas. It means that an increase in the temperature of the gas will increase the speed of the molecules and eventually increase the root mean square of the mean average speed.