where R=8.314 J/(mol·K) and ℳ is molar mass in kilograms per mole. Note that a joule is the same as a kg·m2/s2.
What is the rms speed of N2 molecules at 335 K?
What is the rms speed of He atoms at 335 K?
rms = sqrt (3RT/M)
To calculate the root mean square (rms) speed of gas molecules or atoms, we can use the following formula:
v(rms) = √((3kT) / (m))
Where:
- v(rms) is the rms speed of the molecules or atoms.
- k is the Boltzmann constant, which is approximately 8.314 J/(mol·K).
- T is the temperature in Kelvin.
- m is the molar mass in kilograms per mole.
For N2 molecules:
- R = 8.314 J/(mol·K)
- ℳ(N2) = 28.0134 g/mol = 0.0280134 kg/mol (molar mass of N2)
First, convert the molar mass of N2 from grams to kilograms:
ℳ(N2) = 0.0280134 kg/mol
Next, plug the values into the formula:
v(rms) = √((3 * 8.314 J/(mol·K) * 335 K) / (0.0280134 kg/mol))
Calculate the rms speed for N2:
v(rms) = √((3 * 8.314 * 335) / 0.0280134) ≈ 515.23 m/s
So, the rms speed of N2 molecules at 335 K is approximately 515.23 m/s.
For He atoms:
- ℳ(He) = 4.0026 g/mol = 0.0040026 kg/mol (molar mass of He)
Convert the molar mass of He from grams to kilograms:
ℳ(He) = 0.0040026 kg/mol
Plug the values into the formula:
v(rms) = √((3 * 8.314 J/(mol·K) * 335 K) / (0.0040026 kg/mol))
Calculate the rms speed for He:
v(rms) = √((3 * 8.314 * 335) / 0.0040026) ≈ 1319.24 m/s
So, the rms speed of He atoms at 335 K is approximately 1319.24 m/s.