Introduction:
In order to understand why the root mean square velocity is equal to the escape velocity from the surface of the earth for hydrogen and oxygen at a certain temperature, we need to consider the kinetic theory of gases and the concept of escape velocity.

Kinetic Theory of Gases:
The kinetic theory of gases states that the average kinetic energy of gas molecules is directly proportional to the temperature of the gas. This means that as the temperature increases, the average kinetic energy and hence the velocity of the gas molecules also increase.

Escape Velocity:
Escape velocity is the minimum velocity an object must have in order to escape the gravitational pull of a planet or other celestial body. It is given by the formula:

Escape velocity = sqrt(2 * gravitational constant * mass of the planet / radius of the planet)

Root Mean Square Velocity:
The root mean square velocity is a measure of the average velocity of gas molecules in a sample. It is given by the formula:

Root mean square velocity = sqrt(3 * Boltzmann constant * temperature / molar mass)

Equating the Root Mean Square Velocity and Escape Velocity:
To find the temperature at which the root mean square velocity is equal to the escape velocity, we need to equate the two formulas:

sqrt(3 * Boltzmann constant * temperature_H2 / molar mass_H2) = sqrt(2 * gravitational constant * mass_earth / radius_earth)
sqrt(3 * Boltzmann constant * temperature_O2 / molar mass_O2) = sqrt(2 * gravitational constant * mass_earth / radius_earth)

Simplifying these equations, we get:

3 * Boltzmann constant * temperature_H2 / molar mass_H2 = 2 * gravitational constant * mass_earth / radius_earth
3 * Boltzmann constant * temperature_O2 / molar mass_O2 = 2 * gravitational constant * mass_earth / radius_earth

Since the molar mass of hydrogen (H2) is smaller than the molar mass of oxygen (O2), it takes a higher temperature for hydrogen molecules to have the same root mean square velocity as oxygen molecules.

Calculating the Temperature:
To calculate the temperature at which the root mean square velocity is equal to the escape velocity, we need to rearrange the equations and solve for temperature:

temperature_H2 = (2 * gravitational constant * mass_earth * molar mass_H2) / (3 * Boltzmann constant * radius_earth)
temperature_O2 = (2 * gravitational constant * mass_earth * molar mass_O2) / (3 * Boltzmann constant * radius_earth)

Substituting the known values for the gravitational constant, mass of the earth, molar masses of hydrogen and oxygen, and the radius of the earth, we can calculate the temperatures:

temperature_H2 = (2 * 6.67430 × 10^-11 m^3 kg^-1 s^-2 * 5.972 × 10^24 kg * 2.016 g/mol) / (3 * 1.38064852 × 10^-23 J/K * 6.371 × 10^6 m)
temperature_O2 = (2 * 6.67430 × 10^-11 m^3 kg^-1 s^-2 * 5.972 ×