Question

The temperature of a gas is raised from 27 $^{o}C$ to 927 $^{o}C$. The root mean square speed
(A) is $\sqrt{\dfrac{927}{27}} $ times the earlier value
(B) Remains the same
(C) Gets halved
(D) Gets doubled

Answer 1

Hint: To solve this question we should know the equation for root mean square velocity. The root mean square velocity is a very important term in energy calculation for gases.

Complete step by step solution:
Given in the question:
Temperature of the gas is raised
Initial temperature of the gas ${{T}_{1}}$= 27 $^{o}C$= 300K
Final temperature of the gas ${{T}_{2}}$= 927 $^{o}C$= 1200 K
The formula for the calculation of root mean square velocity is:
\[{{V}_{rms}}=\sqrt{\dfrac{3kT}{m}}\]
From this equation we can notice that the root mean square velocity is proportional to the square root of temperature so we can write:
\[\begin{align}
& \dfrac{{{V}_{rms1}}}{{{V}_{rms2}}}=\sqrt{\dfrac{{{T}_{1}}}{{{T}_{2}}}} \\
& \Rightarrow \dfrac{{{V}_{rms1}}}{{{V}_{rms2}}}=\sqrt{\dfrac{300}{1200}} \\
& \Rightarrow \dfrac{{{V}_{rms1}}}{{{V}_{rms2}}}=\dfrac{1}{2} \\
\end{align}\]
$ \Rightarrow {{V}_{rms2}}=2({{V}_{rms1}})$

Hence the correct answer is option (D)

Additional information:
In kinetic theory of gases there are several types of velocities like the average speed or the mean speed, the most probable velocity or the root mean square velocity. Each velocity has its own importance and thus none of them can be neglected. Mean speed is the average speed of all the molecules irrespective of their direction of motion. Whereas the most probable speed is the speed whose particle in number is maximum which means that in the category of velocity we will find the maximum number of particles.

Note: The root mean square is a very important term in the energy calculation for the gases. It directly corresponds to the energy of the sample unlike the mean speed and the most probable speed. The chances of mistakes anyone could make is that they get confused while writing the formula.