Question

The molecular velocity of any gas:
  (A) is proportional to absolute temperature.
  (B) is proportional to the square of absolute temperature.
  (C) is proportional to the square root of absolute temperature
  (D) is independent of temperature

Answer 1

Hint: From the kinetic theory of gases, we would be able to derive root mean square velocity. By examining the root mean square speed equation, a relationship between temperature and velocity of the gas molecules can be obtained.

Complete step by step answer:
- Let's start with the idea of the kinetic theory of gases. According to the theory, all gaseous particles are in constant random motions at temperatures above absolute zero. By taking both speed and direction into account, velocity is used to describe the movement of gas particles.
- We cannot find the velocity of individual particles and hence we measure their average behavior. Since the particles are moving in random directions, we need to overcome this directional component of velocity.
- Hence, we square the velocities and then its square root is taken. This gives us the root mean square velocity. It is the measure of the speed of particles in gas and can be defined as the square root of the average velocity-squared of molecules in a gas. It can be represented as

   ${{\nu }_{rms}}=\sqrt{\dfrac{3RT}{M}}$
 Where ${{\nu }_{rms}}$ is the root mean square velocity
               R is the molar gas constant
               T is the temperature in kelvin
               M is the molar mass of gas in $kgmo{{l}^{-1}}$

 -Here root mean square speed takes both temperature and molecular weight into account and both these factors directly affect the kinetic energy of the molecule.
- As we mentioned, when we examine the root mean square speed equation, we can see that the changes in temperature (T) and molar mass (M) affect the speed of the gas molecules. From the above rms equation, it's clear that the speed of the molecules in a gas is proportional to the temperature and is inversely proportional to molar mass of the gas. Or we can say that, as the temperature of a sample of gas is increased, the molecules speed up and the root mean square molecular speed increases as a result. So, the correct answer is “Option C”.

Note: Do not confuse rms value with average velocity and most probable velocity. As we have seen rms value is the square root of mean square speed(${{\nu }_{rms}}=\sqrt{\dfrac{3RT}{M}}$) whereas average velocity is the arithmetic mean of speed of all molecules in a gas( $\sqrt{\dfrac{8RT}{\pi M}}$) and most probable velocity is the velocity possessed by maximum number of molecules at any given temperature.( $\sqrt{\dfrac{2RT}{M}}$)