Two ideal gases,X and Y,are thoroughly mixed and at thermal equilibrium in a single container.The molecular mass of X is 4 times that of Y.What is the ratio of root-mean-square velocities of the two gases,v_X,rms /v_Y,rms?

Question

Quizplus · Accepted Answer

The root-mean-square velocity of an ideal gas is given by

v_rms = √(3kT/m)

where k is the Boltzmann constant, T is the temperature, and m is the mass of the gas molecule.

Since the gases are at thermal equilibrium, they must have the same temperature. Let this temperature be denoted by T.

Let the molecular mass of gas X be denoted by M_X, and the molecular mass of gas Y be denoted by M_Y.

Thus, for gas X,

v_X,rms = √(3kT/M_X)

and for gas Y,

v_Y,rms = √(3kT/M_Y)

Taking the ratio of these two velocities, we get

v_X,rms / v_Y,rms = √(M_Y/M_X)

Substituting the given masses, we get

v_X,rms / v_Y,rms = √(1/4) = 1/2

Therefore, the ratio of root-mean-square velocities of gas X to gas Y is 1/2, which corresponds to choice C).

Two Ideal Gases,X and Y,are Thoroughly Mixed and at Thermal