Question

If the RMS velocity of hydrogen becomes equal to the escape velocity from the earth surfaces, then the temperature of hydrogen gas would be:
A) $1.6 \times {10^5}K$
B) $50300K$
C) $82700K$
D) ${10^4}K$

Answer 1

Hint: Escape velocity is the minimum velocity with which a body should be protected from the surface of the earth so that it can escape from the gravitational field of the earth. The escape velocity on the surface of the earth is 11200m/s. Escape velocity on the surface of the moon is 2370m/s.

Formula used:
To solve this type of question we use the following formula.
The root mean square (RMS) velocity is given below.
${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $, where $R$ is the gas constant, $M$ is mass and $T$ is the temperature.
The escape velocity of earth is $11.2km/s$

Complete step by step answer:
According to the question, the RMS velocity of hydrogen is equal to the escape velocity of the earth.
Let us use the formula ${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $and write the following.
$\sqrt {\dfrac{{3RT}}{M}} = 11.2 \times {10^3}$
Now we have the values, the Gas constant is $R = 8.314$ and
Molecular mass of Hydrogen $M = 2 \times {10^{ - 3}}$
Let us substitute these values,
$\Rightarrow \sqrt {\dfrac{{3 \times 8.314T}}{{2 \times {{10}^{ - 3}}}}} = 11.2 \times {10^3}$
Let us square both the sides.
$\Rightarrow \dfrac{{3 \times 8.314T}}{{2 \times {{10}^{ - 3}}}} = {\left( {11.2 \times {{10}^3}} \right)^2}$
Let us simplify it.
$\Rightarrow \dfrac{{24.942T}}{{2 \times {{10}^{ - 3}}}} = 125.44 \times {10^6}$
Let us further simplify.
$\Rightarrow 24.942T = 250.88 \times {10^3} \Rightarrow T = \dfrac{{250.88 \times {{10}^3}}}{{24.942}} = 10.06 \times {10^3}$
We can also write it as below.
$\Rightarrow T = {10^4}K$

$\therefore $ The temperature of hydrogen gas would be ${10^4}K$. Hence, option (D) correct.

Note:
- The relation between escape velocity and orbital velocity is given below.
Escape velocity= $\sqrt 2 \times $ orbital velocity.
- The root means square velocity is the square root of the average of the square of the velocity and it has units of velocity.
- The RMS velocity is used instead of the average because for a typical gas sample the net velocity is zero since the particles are moving in all directions. This is an important formula as the velocity of the particles determines both the diffusion and effusion rates.