Root-mean-square speed, often abbreviated as \( v_{rms} \), is a mathematical way of representing the speed of gas particles in a container. It accounts for the fact that particles in a gas don't move uniformly but cover a range of speeds.
The formula for calculating the root-mean-square speed is \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass. Of the three gases we discussed, neon, krypton, and radon, the speed differences arise from their varying molar masses.
Since Neon has the smallest molar mass, it ends up having the highest \( v_{rms} \), meaning its particles move the fastest. On the contrary, Radon, with the largest molar mass, has the slowest particles.
- High molar mass (Rn): slower speeds
- Low molar mass (Ne): faster speeds
This shows how inversely the root-mean-square speed of particles relates to the square root of their molar mass, making each gas particle's speed distinct.