Theoretical Foundation
The root mean square velocity (vrms) is derived from the Maxwell-Boltzmann distribution of molecular velocities in an ideal gas. It represents the square root of the mean squared velocity of gas molecules.
Mathematical Definition:
\[v_{rms} = \sqrt{\frac{1}{N}\sum_{i=1}^{N} v_i^2}\]
For an ideal gas, this simplifies to:
\[v_{rms} = \sqrt{\frac{3RT}{M}}\]
Key Relationships:
\[E_k = \frac{1}{2}mv^2 = \frac{3}{2}kT\]
\[PV = nRT = NkT\]
Where:
\[
\begin{cases}
R = 8.314 \text{ J/(mol·K)} & \text{(Gas constant)} \\
T = \text{Temperature in Kelvin (K)} \\
M = \text{Molar mass in kg/mol} \\
k = 1.380649 × 10^{-23} \text{ J/K} & \text{(Boltzmann constant)}
\end{cases}
\]