A sealed container contains 1 mole of neon gas at STP. Estimate the number of neon atoms having speeds in the range from 200 m/s to 202 m/s. (Hint: Assume the probability of neon atoms having speeds between 200 m/s and 202 m/s is constant).

Answer 1

STP - Standard Temperature and Pressure - is 273 and 105 Pa
Maxwell distribution for the relative velocities is
ΔN/N = 4/sqrt(π)•(1/e^(u^2))•u^2•Δu,
v1 = 200m/s, v2 =202 m/s, Δv =202-200 = 2m/s.
The most probable velocity
v(pr) = sqrt(2RT/M) = sqrt(2•8.31•273/20•10^-3) = 476.3 m/s.
The relative velocity is
u = v1/v(pr) = 200/476.3 = 0.42,
u^2 = 0.176,
1/e^(u^2)) = 0.838
Δu = Δv/v(pr) = 2/476.3 = 4.2•10^-3.
ΔN/N = 4/sqrt(π)•0.838•0.176•4.2•10^-3 =1.4•10^-3 = 0.0014.
Therefore. 1,4% of the neon atoms have the velocities of 200 – 202 m/s or their absolute number in 1 mole is
ΔN = N•0.0014 = 6.022•10^23•0.0014 = 8.42•10^20

Answer 2

This looked a little sloppy imo... Here's my example

1. Find most probably velocity of Krypton
v = √[(2*8.314J/K*273K)/(0.083798 kg)] = 237.75 m/s

Find the ratio of each velocity
205/237.75 = .8808 The average is .8851
207/237.75 = .8894

Find the ratio of change in velocities 207-205
2/237.75 = 0.008593

Then: Find the probability!
6.022e23 molecules * 5 mol * .8851 * 0.008593 = 2.29e22

I just did a problem like this... These are my numbers, but the work is a lot more simple