Question

A planet with radius R and acceleration due to gravity g, will have atmosphere only if r.m.s. speed of air molecules is less than

• A. $1.414 \sqrt{gR}$
• B. $ 1.732 \sqrt{gR}$
• C. $2 \sqrt{gR} $
• D. $ 3.14 \sqrt{gR} $
• E. $ 2.75 \sqrt{gR} $

Answer 1

The Correct Option is A

The r.m.s. speed of air molecules is the square root of the average velocity squared. This is given by the equation: 

\[ v_{\text{r.m.s.}} = \sqrt{\frac{3kT}{m}} \] Where: - \( k \) is the Boltzmann constant, - \( T \) is the temperature of the atmosphere, - \( m \) is the mass of an air molecule. The atmosphere of a planet will remain intact as long as the r.m.s. speed of air molecules is less than the escape velocity, which is the velocity required for a molecule to escape the planet's gravity. The escape velocity \( v_{\text{esc}} \) is given by: \[ v_{\text{esc}} = \sqrt{2gR} \] Where: - \( g \) is the acceleration due to gravity, - \( R \) is the radius of the planet. For the atmosphere to remain, the r.m.s. speed of air molecules must be less than the escape velocity. Therefore, the condition for the atmosphere to exist is: \[ v_{\text{r.m.s.}} < v_{\text{esc}} \] Substituting the value for escape velocity: \[ \sqrt{\frac{3kT}{m}} < \sqrt{2gR} \] Simplifying the expression: \[ T < \frac{2gR}{3k} \] In this context, the value given is approximately \( 1.414 \sqrt{gR} \), meaning that for the planet to have an atmosphere, the r.m.s. speed must be less than this value.

Correct Answer:

Correct Answer: (A) \( 1.414 \sqrt{gR} \)