Question

Given is 1mole of nitrogen molecules at atmospheric pressure and temperature of 20deg C.

a) Find the number of molecules having their speed between 200m/s and 201m/s

b) Find the most probable speed at this temperature

c) Find the number of molecules with the most probable speed (within 1 m/s form it)

d) Find the number of molecules with the speed of 1000m/s (within 1 m/s form it)

e) Find the number of molecules with the speed of 2000m/s (within 1 m/s form it)

Answer 1

1 mole of Nitrogen contains N= 6x 10^23 molecules (Avgadro's number)

Molecular weight of Nitrogen M₀ = 0.014 kg

R = 8.314 gas constant

The number of molecule with speed v is given by

dN = 4piN(M₀/2piRT)^3/2 v² e^{-M0v^2/2RT} dv

substituting the constant values we will have

dN = 6.59746 x 10^15 x v² x e^(-2.878x10^-6)v^2 x dv, the exponent being very small cna be approximated to 1

      = 6.59746 x 10^15 xV^2 x dv

a) v= 200 and dv =1 dN = 26.36 x 10^19

b) most probable speed = SQRT(2*8.314*293/0.014) = 590 m/s

c) v= 590 and dv =1 dN= 3.48 x 10^20

d) v= 1000 and dv =1,now the exponent value becomes significant = 0.056

                       dN = 6.6 x10^15 x 1000^2 x 0.056 = 3.69 x 10^20

e) v= 2000, dv = 1 exponent value = 10^-5

                       dN = 6.6 x 10^15 x 2000 x2000 x 10^-5 = 26.4 x 10^16