Question

The root mean square speed of a gas of density 1.5 g/ litre at a pressure of is $2\times {10^6}~{N}{m^{-2}}$?

Answer 1

Hint: Gases are composed of atoms or molecules that flow in random directions at variable rates. The root mean square velocity (RMS velocity) is a method for calculating a single velocity for the particles. The root mean square velocity formula is used to calculate the mean velocity of a gas particle.

Complete answer:
We can see from the root mean square speed formula that changes in molar mass and the temperature impact the pace of gas molecules. The speed of molecules in a gas is related to temperature and inversely related to the gas's molar mass. In other words, when the temperature of a gas sample rises, the molecules accelerate, and the root mean square molecular speed rises as well.

In the given question, we need to find out the root mean square speed, when density of the gas is equal to $1.5g/litre$ and the pressure of the gas is $2\times {{10}^{6}}N/{{m}^{2}}$

The formula of the root mean square velocity can be written as:
$\overline{c}=\sqrt{\dfrac{3P}{\rho }}$
Where $P$ is the pressure of the gas and $\rho $is the density of the gas.
$\Rightarrow \sqrt{\dfrac{3\times 2\times {{10}^{6}}}{1.5}}$
$\Rightarrow 2\times {{10}^{3}}m/s$

Hence the root mean square speed of the given is $2\times {{10}^{3}}m/s$.

Note: The RMS computation yields the root mean square speed rather than the velocity. This is due to the fact that velocity is a vector quantity with magnitude and direction and speed is the scalar quantity with only magnitude. The RMS computation provides simply the magnitude or speed.