Molecular speeds: A sealed container holds 0.020 moles of nitrogen (N₂) gas at a pressure of 1.5 atmospheres and a temperature of 290 K. The atomic mass of nitrogen is 14 g/mol. The Boltzmann constant is 1.38 × 10^-23 J/K and the ideal gas constant is R = 8.314 J/mol ∙ K = 0.0821 L ∙ atm/mol ∙ K. The average translational kinetic energy of a nitrogen molecule is closest to A) 4.0 × 10^-21 J. B) 6.0 × 10^-21 J. C) 8.0 × 10^-21 J. D) 10 × 10^-21 J. E) 12 × 10^-21 J.

Question

Quizplus · Accepted Answer

First, we can use the ideal gas law to find the volume of the container:

PV = nRT

V = (nRT)/P 
V = (0.020 mol)(0.0821 L∙atm/mol∙K)(290 K)/(1.5 atm) 
V ≈ 1.06 L

Next, we can calculate the root-mean-square velocity (vrms) of the nitrogen molecules using the formula:

vrms = sqrt((3kT)/(m))

where k is the Boltzmann constant, T is the temperature in kelvin, and m is the mass of one nitrogen molecule.

First, we need to find the mass of one nitrogen molecule:

mass = (14 g/mol)/(6.02×10^23 molecules/mol) 
mass ≈ 2.33×10^-23 g/molecule

Then, we can plug in the given values:

vrms = sqrt((3(1.38×10^-23 J/K)(290 K))/(2.33×10^-23 kg)) 
vrms ≈ 517 m/s

Finally, we can use the formula for kinetic energy of a molecule:

KE = (1/2)mv^2

KE = (1/2)(2.33×10^-23 kg)(517 m/s)^2 
KE ≈ 6.0×10^-21 J

Therefore, the average translational kinetic energy of a nitrogen molecule is closest to 6.0×10^-21 J, or choice B.

Molecular Speeds: a Sealed Container Holds 0