Kinetic Theory — Practice

Question 1

2 moles of an ideal gas are enclosed in a container of volume 0.05 m^3 at 300 K. Find the pressure (R = 8.314 J/mol/K).

Accepted Answer

P = mu R T / V = (2 x 8.314 x 300) / 0.05 = 4988.4 / 0.05 = 9.98 x 10^4 Pa.

Question 2

Find the rms speed of oxygen molecules at 300 K (molar mass 0.032 kg/mol, R = 8.314 J/mol/K).

Accepted Answer

v_rms = sqrt(3RT/M) = sqrt(3 x 8.314 x 300 / 0.032) = sqrt(233831) = 484 m/s.

Question 3

Using the law of equipartition of energy, estimate the molar specific heat of a solid element at room temperature.

Accepted Answer

C = 3R = 3 x 8.314 = 24.9 J/mol/K (the Dulong-Petit law), from each of N atoms vibrating in 3 dimensions contributing 3 kB T per atom.

Question 4

A gas has number density 3 x 10^25 per m^3, molecular mass 4.65 x 10^-26 kg, and mean square speed 2.66 x 10^5 m^2/s^2. Find its pressure.

Accepted Answer

P = (1/3) n m (v-squared average) = (1/3) x 3x10^25 x 4.65x10^-26 x 2.66x10^5 = 1.24 x 10^5 Pa.

Question 5

Using the law of equipartition of energy, find Cv, Cp, and gamma for a rigid diatomic gas.

Accepted Answer

5 degrees of freedom (3 translational + 2 rotational). Cv = 5R/2, Cp = Cv + R = 7R/2, gamma = Cp/Cv = 7/5 = 1.4.

Question 6

Estimate the mean free path of a gas with number density 2.7 x 10^25 per m^3 and molecular diameter 2 x 10^-10 m.

Accepted Answer

l = 1/(sqrt(2) n pi d^2) = 1/(1.4142 x 2.7x10^25 x 3.1416 x 4x10^-20) = 1/(4.80x10^6) = 2.08 x 10^-7 m.

Question 7

A vessel contains a mixture of 2 mol of oxygen and 3 mol of nitrogen at a total pressure of 5 x 10^5 Pa. Find the partial pressure of each gas.

Accepted Answer

Mole fractions: O2 = 2/5 = 0.4, N2 = 3/5 = 0.6. Partial pressures: P(O2) = 0.4 x 5x10^5 = 2 x 10^5 Pa, P(N2) = 0.6 x 5x10^5 = 3 x 10^5 Pa.

Question 8

A gas at pressure 2 x 10^5 Pa, volume 4 L, and temperature 300 K is compressed to 2 L at 350 K. Find the new pressure.

Accepted Answer

P1V1/T1 = P2V2/T2, so P2 = P1(V1/V2)(T2/T1) = 2x10^5 x (4/2) x (350/300) = 4.67 x 10^5 Pa.

Question 9

A polyatomic gas molecule has 3 translational, 3 rotational, and 2 active vibrational modes. Find Cv, Cp, and gamma.

Accepted Answer

Using Cv = (3 + f)R with f = 2 vibrational modes: Cv = 5R. Cp = Cv + R = 6R. gamma = Cp/Cv = 6/5 = 1.2.

Question 10

Hydrogen (M = 2 u) and oxygen (M = 32 u) are at the same temperature. Find (a) the ratio of their rms speeds, and (b) the temperature at which oxygen's rms speed at that temperature equals hydrogen's rms speed at 300 K.

Accepted Answer

(a) v_rms ratio = sqrt(M(O2)/M(H2)) = sqrt(32/2) = 4, so hydrogen is 4 times faster at the same T. (b) Setting 3RT2/32 = 3R(300)/2 gives T2 = 300 x (32/2) = 4800 K.

Practice — Kinetic Theory

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