What is $v_{rms}$? Write the equation of $v_{rms}$ in terms of: $(1)$ density,$(2)$ mass of a molecule,$(3)$ molar mass.

(N/A) $v_{rms}$ stands for the root mean square speed of gas molecules. It is defined as the square root of the mean of the squares of the speeds of individual molecules in a gas.
$(1)$ In terms of density $(\rho)$ and pressure $(P)$: $v_{rms} = \sqrt{\frac{3P}{\rho}}$
$(2)$ In terms of mass of a molecule $(m)$: $v_{rms} = \sqrt{\frac{3k_BT}{m}}$,where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.
$(3)$ In terms of molar mass $(M)$: $v_{rms} = \sqrt{\frac{3RT}{M}}$,where $R$ is the universal gas constant and $T$ is the absolute temperature.