Air,which is primarily nitrogen and oxygen,does contain a very small fraction of helium.What is the root-mean-square speed of helium at 25 $\circ$ C,where its average kinetic energy is about 3.72 kJ/mol? How many times faster or slower does it travel than N₂? (1 J = 1 kg .m²/s²; 1 mol = 6.02 $\times$ 10²³)

Question

Quizplus · Accepted Answer

Using the equation for the root-mean-square speed of a gas molecule:

u_rms = sqrt((3RT)/(M))

where R is the gas constant, T is the temperature, and M is the molar mass. For helium at 25°C (298 K), using the molar mass of helium (4.003 g/mol) and the given kinetic energy (3.72 kJ/mol):

3.72 kJ/mol = (1/2)MV^2
V^2 = (2*3.72 kJ/mol)/(4.003 g/mol)
V^2 = 1.848 km^2/s^2
V = sqrt(1.848) km/s
V = 1.358 km/s

Converting to m/s:

u_rms = 1358 m/s

To find how many times faster or slower this is than nitrogen gas:

u_rms,N2 = sqrt((3RT)/(28 g/mol))

At 25°C, this gives:

u_rms,N2 = 515 m/s

Taking the ratio of the two:

u_rms,He/u_rms,N2 = 1358 m/s / 515 m/s = 2.64

Therefore, helium travels about 2.65 times faster than nitrogen gas at 25°C.

The correct answer is A.

Air,which Is Primarily Nitrogen and Oxygen,does Contain a Very Small ∘\circ∘

Air,which Is Primarily Nitrogen and Oxygen,does Contain a Very Small $\circ$