What is $rms$ value? What is $v_{rms}$? Derive the equation of $v_{rms}$ in terms of pressure.

(N/A) The $rms$ (root mean square) value is defined as the square root of the mean of the squares of the values.
$v_{rms}$ is the square root of the mean of the squares of the speeds of gas molecules.
According to the kinetic theory of gases,the pressure $P$ of an ideal gas is given by:
$P = \frac{1}{3} \rho \langle v^2 \rangle$
where $\rho$ is the density of the gas and $\langle v^2 \rangle$ is the mean square speed.
Rearranging the equation for $\langle v^2 \rangle$:
$\langle v^2 \rangle = \frac{3P}{\rho}$
Since $v_{rms} = \sqrt{\langle v^2 \rangle}$,we substitute the expression for $\langle v^2 \rangle$:
$v_{rms} = \sqrt{\frac{3P}{\rho}}$