Question

For N2 gas at 1 atm and 15ºC, calculate: a)the number of collisions each molecule encounter in 1s b)the total number of collisions in a volume of 1 m3 in 1 s c) the average distance the N2 molecules travel between collisions.

Answer 1

a)

No. coliisions per molecule per second:

Collision frequency = RMS / MFP

where

RMS = Root-mean-square velocity

MFP = mean free path

claculate each:

RMS

RMS =sqrt(3*RT/MW)

where, R = ideal gas constant, T = temperature, MW moalr mas in kg/mol

MW of N2 = 28.08 g/mol --> change to kg --> 28.08 *10^-3 kg/mol

T = 15°C = 15+273.15 K = 288.15 K

RMS= sqrt(3*8.314*288.15 / (28.08 *10^-3))

RMS = 505.913 m/s

Now,

MFP

Assume molecules have diameter "D"; when, for a circle, then, the Area (σ)

σ = d = 3.75*10^-10 m

MFP = RT/((sqrt(2) * PI* σ^2 *Na*P))

where,

R = 8.314 J/molK, ideal gas constant,

T = Abs. Temperature = 15°C = 15+273.15 K = 288.15 K

σ = area of collision

Na = avogrados' number = 6.022*10^23

P = pressure abs = 1 atm = 101325 Pa

Now;

get

d = 370 pm

σ = 2*d = 2*370 pm = 740 pm = 740*10^-12 m

substitute all data:

MFP = RT/((sqrt(2) *PI * σ^2 *Na*P))

MFP = (8.314)(288.15 ) / (sqrt(2) * 3.1416* (740*10^-12)^2 * (6.022*10^23)(101325))

MFP = 2395.6791 /148452217468

MFP = 1.613 *10^-8 m

Now,

substitute in Collisions per frequency

Collision frequency = RMS / MFP

Collision frequency = 505.913 m/s / 1.613 *10^-8 m

Collision = (505.913 )/(1.613 *10^-8) = 31364724116.6 1/s = 3.13*10^10 Hz

b)

total no. collision in V = 1 m3 in 1 s

Total Number of Collision per unit volume per second...

Total Collision = 1/2*n' * Collisions per second;

where n' = N/V = molecules per volume

n' = N/V = (Na*P)/(R*T) = (6.022*10^23)(101325)/(8.314*288)

n' = 2.5483*10^25

Total Collision = 1/2*n' * Collisions per second;

Total Collision = 1/2*(2.5483*10^25) * (3.13*10^10) = 3.988*10^35 per m3 per second

c)

Averge distance between N2 molecules

This is by definition, the mean free path between N2

MFP = 1.613 *10^-8 m

MFP = 1.613 *10^-8 m * 10^9 nm / m = 16.13 nm