Question

Explain what happens to the temperature when the root mean square velocity is doubled?
A. Increased by four times
B. Increased by two times
C. Reduced to a half
D. Reduce to a fourth

Answer 1

Hint : The name of the concept actually offers a pretty good hint as to what the root mean square velocity is as it is said to be directly proportional to the square root of the quotient of the temperature and molecular mass. With this in mind, try to answer the given question.

Complete step by step solution : Let us first look into what the root mean square velocity of a particle really is before trying to evaluate the formula which can be used to calculate it.
Although the velocity of gaseous particles is constantly changing, the distribution of velocities does not change. It is very difficult to judge the velocity of each individual particle, so we often express these velocities in terms of the particles’ average behaviour which is why particles moving in opposite directions have velocities of opposite signs. Now since the particles constituting a gas are in random motion, it is very possible that there will be approximately as many moving in one direction as in the opposite, meaning that the average velocity for a collection of gas particles equals zero. That makes this value completely useless, therefore we need a different method to help calculate the average of velocities.
Now, by squaring the velocities and taking the square root, we overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity. Since the value excludes the particles’ direction, we now refer to the value as the average speed. The root-mean-square speed 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. The final equation obtained is:

${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$

Thus, we conclude that the root mean square of velocity of a particle is proportional to $\sqrt{\dfrac{T}{M}}$. Therefore, by this relation we can easily conclude that the root mean square velocity of a particle will double only when its temperature is quadrupled.
Therefore, we can conclude that the answer to this question is ‘A. Increased by four times’

Note : The derivation of the exact value of the root mean square velocity of a particle in a system instead of a logical breakdown can be done from the formula of the kinetic energy of gases. For more information, it is advisable to read through the entire kinetic theory of gases one more time.