Homework #6 (chapter 22): Molecular Properties of Gases

Exercise 4:

a) pV=NkT è N/V = p/(kT) = 105 (N/m2)/(1.38*10-23 (Nm/K) *285K) = 2.54 * 1025 m-3

(N/V)1/3 = 2.94*108 / m è Intermolecular distance between adjacent molecules is 3.40 * 10-9 m. This distance is about 11 molecular diameters (3.40 * 10-9 m / 3.00 * 10-10 m » 11).

b) If 1 mol of water has a volume of 18 cm3, then N/V = 6.02*1023 / 1.8*10-5 m3 = 3.34*1028 m-3

(N/V)1/3 = 3.22*109 / m è Intermolecular distance between adjacent molecules is 3.11 * 10-10 m. This distance is about 1 molecular diameter (3.11 * 10-10 m / 3.00 * 10-10 m » 1).

 

Exercise 5:

N/V = p/(kT) = 1.22*1.01*105 (N/m2) /(1.38*10-23 (Nm/K) *308K) = 2.90 * 1025 m-3

Volume occupied by one Ar atom: VAr = 4p/3 * (7.1*10-11 m)3 = 1.50*10-30m3

Volume fraction occupied by Ar atoms in sample: N/V * VAr = 2.90 * 1025 m-3 * 1.50*10-30m3 = 4.35*10-5, i. e. only about four thousandth of one percent of the container volume is occupied by the volume of the Ar atoms.

 

Exercise 6:

Each H2 molecule has a momentum p=mv. When it strikes the wall at an angle of 55°, the momentum transfer per elastic collision is 2 p cosq = 2 m v cosq, accounting for the fact that only the normal component of the force exerts pressure. The force exerted on the wall is

F = dp / dt = dn/dt 2 m v cos q  with dn/dt the collision rate.

p = F / A = (dn/dt 2 m v cos q ) / A = (1.6*1023 * 2 * 3.3*10-27 * 103 * cos55°) / (2 * 10-4)   (kg*m/s*s*m2) = 3.03 Pa

 

Exercise 23:

a) vesc,earth = 1.12*104 m/s (see Table 14-2, p. 310)

vrms = (3kT/m)1/2

H2: vesc,earth = vrms = [3kTesc,earth,H2/m(H2)]1/2 ==> Tesc,earth,H2 = vesc,earth2 * m(H2) / (3k) = (1.12*104)2 m2/s2 * 3.3*10-27 kg / (3*1.38*10-23) kg*m2/(s2*K) = 1.00*104 K

O2: Tesc,earth,O2/Tesc,earth,H2 = mO2/mH2 = 16 ==> Tesc,earth,O2 = 1.6*105 K

b) vesc,moon = 2.38*103 m/s, so vesc,moon / vesc,earth = 0.213

Tesc,moon,H2 = Tesc,earth,H2 * (vesc,moon / vesc,earth)2 = 0.2132 *104 K = 454 K

Tesc,moon,O2 = Tesc,earth,O2 * (vesc,moon / vesc,earth)2 = 0.2132 * 1.6*105 K = 7260  K

c) At 1000 K neither much H2 escapes (requires 10000 K) nor much O2 (requires 160000 K).