QJEEM23Ph5e61: JEE Main 2023 Physics - Class 11 JEEM23Ph5e61 Solution: The rms speed of oxygen molecule in a vessel at particular…

The relationship between the root-mean-square (rms) speed ( $v_{rms}$ ) and the average speed ( $v_{avg}$ ) of molecules in a gas can be found using the Maxwell-Boltzmann distribution. The rms speed and average speed are related as follows:

Where:

• $R$ is the ideal gas constant
• $T$ is the temperature in Kelvin
• $M$ is the molar mass of the gas
• $\pi$ is the mathematical constant pi

In this problem, the rms speed of the oxygen molecule is given by:

Now, let's divide the expression for $v_{rms}$ by the expression for $v_{avg}$ :

By simplifying the expression, we get:

Square both sides of the equation:

Now we will substitute the provided value of $\pi = \frac{22}{7}$ :

By simplifying the expression, we get:

Now let's solve for $x$ :

Multiplying both sides by $x$ :

Finally, solving for $x$ :

So, the value of $x$ is $\boxed{28}$ .