Find numerical answers to the following questions at room pressure and room temp

Question

Find numerical answers to the following questions at room pressure and room temperature: (a) What is the average density n (number of molecules/cm) of air (N_2)? (b) What is the average speed v of the N_2 molecule of a mass M? (c) What is the mean free path of the N_2 molecule when the scattering rate is given by Gamma = nov, where sigma is the classical cross section of the N_2 molecule? (d) What is the quantum concentration n_q = (Mk_B T/2 pi h^2)^3/2, of air (N_2)? (e) Compare the above two densities and determine whether the system is in classical or quantum regime.

Explanation / Answer

a. 1 mole of N2 at STP is 22.4 l

so, n = 1 mole per 22400 cm^3 = 6.022*10^23/22400 = 2.6883*10^19 molecules per cm^3

b. Average velocity of a gas molecule, v = sqroot(8*kb*T/pi*M) where M is mass of one molecule of gas and kb is noltzmann's constant

v = sqroot(8*1.38*10^-23*300/pi*28*1.6*10^-27) = 485.22 m/s

c. Mean free path, lambda = kb*T/pi*d^2*p*sqroot(2)

d = 364*10^-12 m

p = 1 atm = 1.01*10^5 Pa

lambda = 6.966*10^-8 m

d. nq = (Mkb*T/2*pi*h^2)^3/2 [ h is planks constant/2*pi ]

nq = (28*1.6*10^-27 * 1.38*10^-23 * 300 / 2*pi* (6.62*10^-34)^2)^3/2 = 106.3878*10^24

e. hence the nq is too high for the particles to be in quantum state and hence athey are classical regime

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