Question

At what temperature the mean kinetic energy of hydrogen molecules increases to such that they will escape out of the gravitational field of earth forever? (take $ {v_e} = 11.2km{\sec ^{ - 1}} $ )
 $ 1)12075K $
 $ 2)10000K $
 $ 3)20000K $
 $ 4)10075K $

Answer 1

Hint: At a particular temperature, the different molecules of a gas possess different speeds. Further due to continuous collisions of the molecules among themselves their speed keeps on changing. According to the kinetic theory of gases, the average kinetic energy of a gas molecule is directly proportional to absolute temperature.

Complete answer:
The average kinetic energy of a gas molecule is given by the formula;
 $ K.E = \dfrac{3}{2}KT $
Where, K = Boltzmann constant
= $ 1.38 \times {10^{ - 23}}J{K^{ - 1}} $
T = Absolute temperature
Escape velocity $ \left( {{v_e}} \right) $ is given in the question.
 $ {v_e} = 11.2Km{\sec ^{ - 1}} $
= $ 11.2 \times {10^3}m{\sec ^{ - 1}} $
In order to escape the gravitational field of earth forever;
 $ \dfrac{3}{2}KT = \dfrac{1}{2}m{v_e}^2 $
Where, m = mass of hydrogen
= $ 2 \times 1.67 \times {10^{ - 27}}Kg $
Putting all the values in the formula, we can find the value of temperature;
 $ T = \dfrac{{m{v^2}_e}}{{3K}} $
= $ \dfrac{{2 \times 1.67 \times {{10}^{ - 27}}Kg \times {{(11.2 \times {{10}^3}m{{\sec }^{ - 1}})}^2}}}{{3 \times 1.38 \times {{10}^{ - 23}}J{K^{ - 1}}}} $
 $ = 10120K $
Which is closer to option 4
 $ T \simeq 10075K $
The answer is option 4 .

Additional Information:
It should be noted that there are three types of speeds of gaseous molecules that are commonly used. These are; Most probable speed: it is the speed possessed by the maximum fraction of molecules, Root mean square speed: it is the square root of the mean of the squares of the speeds of molecules, Average speed: it is the average of the different speeds of all the molecules.

Note:
We have found the temperature at which the kinetic energy of hydrogen molecules increases to such that they will escape out of the gravitational field by equating the average kinetic energy of gas molecules with escape velocity. Escape velocity is the exact amount of energy needed to escape the gravitational field of earth.