Question

The average velocity of gas molecules is\[400m/sec\]. calculate its \[rms\]velocity at the same temperature.
\[
  A.\;\;\;\;\;432{\text{ }}m/\sec \\
  B.\;\;\;\;\;434{\text{ }}m/\sec \\
  C.\;\;\;\;\;438{\text{ }}m/\sec \\
  D.\;\;\;\;\;440{\text{ }}m/\sec \\
 \]

Answer 1

Hint: First we have to find the values of \[\dfrac{{RT}}{M}\]and then put it into the formula for\[{u_{rms}}\].then only we can calculate the \[rms\] velocity.
Formula used:
\[
  Root{\text{ }}means{\text{ }}square{\text{ }}velocity{\text{ }} = {\text{ }}{u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \\
  Average{\text{ }}velocity{\text{ }} = {\text{ }}{u_a} = \sqrt {\dfrac{{8RT}}{{\pi \times M}}} \\
 \]
Where,
R=gas constant
T=temperature
M= mass of the gas molecules.

Complete step by step solution:
Before answering let's discuss a little bit about \[rms\] velocity, \[rms\] velocity is the root mean square velocity is the square root of the average of the square of the velocity. It has units similar to that velocity. The reason for using \[rms\] velocity rather than simple velocity is that the molecules simply move in all directions and hence the net velocity is zero.
Now, coming back to question we are given the average velocity \[\left( {{u_a}} \right)\] of the gas molecule which is 400 m/sec.
We know \[
  {u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \\
    \\
 \]
The average velocity of gas molecules is\[400m/sec\].
Therefore, \[400 = \sqrt {\dfrac{{{\text{8RT}}}}{{\pi \times {\text{M}}}}} \]
This can be written as $\dfrac{{{\text{RT}}}}{{\text{M}}} = \dfrac{{160000 \times \pi }}{8}$= $20000 \times \pi $
Now, we are having the value of$\dfrac{{RT}}{M}$.
Now, for finding \[rms\]velocity we will use the formula\[
  {u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \\
    \\
 \], so put the value of $\dfrac{{RT}}{M}$
We get,
${U_{rms}} = \sqrt {3 \times 20000 \times \pi } = \sqrt {188495.56} = 434{\text{ }}m/\sec $

So the answer to this question is option B. \[434{\text{ }}m/sec\]

Note: We must know that the gas molecules are having velocity in every direction it is not possible to calculate the average velocity as it will come out to be zero. For calculating the velocity of every molecule we take \[rms\]velocity into consideration.