Question

At what temperature will the rms speed of oxygen molecules become just sufficient for escaping from the Earth's atmosphere? (Given: Mass of oxygen (m) = $2.76 \times {10^{ - 26}}kg$ Boltzmann's constant ${k_B} = 1.38 \times {10^{ - 23}}J{K^{ - 1}}$)
A.) $5.016 \times {10^4}K$
B.) $2.508 \times {10^4}K$
C.) $1.254 \times {10^4}K$
D.) $8.360 \times {10^4}K$

Answer 1

Hint: In this question rms is root-mean-square speed which is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.

Complete step-by-step answer:

Let at temperature T rms speed of oxygen molecules become just sufficient for escaping from the Earth’s atmosphere

Since, ${V_{escape}} = 11.2km/s$
R.M.S. = $\sqrt {\dfrac{{3KT}}{M}}$
So, ${V_{rms}} = {V_{escape}} = \sqrt {\dfrac{{3KT}}{M}} = 11200{\text{ }}m/s$
$T = \dfrac{{{{11200}^2} \times 2.76 \times {{10}^{ - 26}}}}{{3 \times 1.38 \times {{10}^{ - 23}}}}$
$T = 8.360 \times {10^4}K$

Additional Information:
The Boltzmann Constant is used in numerous physics disciplines. Several are listed below-
A. Boltzmann Constant is used in classical statistical mechanics to describe equipartition of the energy of an atom.
B. It is used to express the element Boltzmann.
C. In the statistical definition of entropy, it plays an important role.
D. It is used in the physics of semiconductors to express thermal tension.

Molecular oxygen is a diatomic molecule made up of two oxygen atoms which are held together by a covalent bond. Molecular oxygen is essential to survival, since it is used by many species for respiration. It is essential for the combustion of fossil fuel, too.

Note: In this question, The Boltzmann constant is the proportionality factor which relates the relative average kinetic energy of particles in a gas to the gas's thermodynamic temperature. This exists in the concepts of the kelvin and the steady flame, and Planck's black-body radiation law and Boltzmann 's entropy formula. The Boltzmann constant has the energy dimension divided by temperature, equal to that of entropy.