The average kinetic energy of the molecules in a gas sample depends only on the temperature, 𝑇. However, given the same kinetic energies, a lighter molecule will move faster than a heavier molecule, as shown in the equation for rms speed

rms speed=3𝑅𝑇ℳ‾‾‾‾‾√ where 𝑅=8.314 J/(mol⋅K)
R=8.314J/(mol⋅K) and ℳ is molar mass in kilograms per mole. Note that a joule is the same as a kilogram‑meter squared per second squared (kg·m2/s2).

What is the rms speed of N2 molecules at 391 K?

rms speed = sqrt (3RT/M). Just substitute the number is the problem.

You're given R, T, and M.

To find the root mean square (rms) speed of N2 molecules at 391 K, we can use the equation:

rms speed = √(3RT/M)

Where:
- R is the gas constant (8.314 J/(mol⋅K))
- T is the temperature in Kelvin (391 K)
- M is the molar mass of the gas in kilograms per mole (N2 = 28.0134 g/mol = 0.0280134 kg/mol)

Let's substitute the values into the equation:

rms speed = √((3 * 8.314 J/(mol⋅K) * 391 K) / 0.0280134 kg/mol)

Calculating the expression within the square root:

rms speed = √(9746.6682 J/(kg·K))

Finally, calculate the square root to get the rms speed:

rms speed ≈ √(9746.6682 J/(kg·K))

Note that the units cancel out, leaving us with the rms speed in meters per second (m/s).