Question

9.) Compute the root-mean-square speed of N2 molecules in a sample of nitrogen gas at a temperature of 206°C.

_____ m s-1
10.) Compute the root-mean-square speed of Ar molecules in a sample of argon gas at a temperature of 78°C.

_____ m s-1
11.) The average molecular speed in a sample of H2 gas at a certain temperature is 1.97×103m/s.

The average molecular speed in a sample of Xe gas is _____ m/s at the same temperature.

12.) A sample of Ne gas is observed to effuse through a pourous barrier in 8.59 minutes. Under the same conditions, the same number of moles of an unknown gas requires 22.1 minutes to effuse through the same barrier.

The molar mass of the unknown gas is ____ g/mol.

Answer 1

The root mean square speed can be calculated if we know the molar mass M and the temperature T in Kelvins:

$$v_\text{rms}=\sqrt{\frac{3RT}{M}}.$$

9) For nitrogen gas, the molecular weight is 0.028 kg/mol, at 206°C, or 479 K, the RMS speed is 653 m/s.

10) For argon with molar mass of 0.04 kg/mol, at 78°C, or 351 K, the RMS speed is 468 m/s.

11) In this problem, first we need to express the temperature in terms of hydrogen gas characteristics:

$$v_\text{rms1}=\sqrt{\frac{3RT}{M_1}}\rightarrow T=\frac{v_\text{rms1}^2M_1}{3R},\\\space\\ v_\text{rms2}=\sqrt{\frac{3RT}{M_2}}=v_\text{rms1}\sqrt{\frac{M_1}{M_2}},\\\space\\ v_\text{rms2}=1.97×10^3\sqrt{\frac{2}{131}}=243\text{ m/s}.$$

12) The rate of effusion of gases depends on time according to the following expression:

$$\frac{n}{t}=\frac{k}{\sqrt{M}}.$$

For neon:

$$n_1=kt_1/\sqrt{M_1},\\ M_2=(kt_2/n)^2,\\\space\\ M_2=M_1\bigg(\frac{t_2}{t_1}\bigg)^2=20\bigg(\frac{22.1}{8.59}\bigg)^2=132\text{ g/mol}.$$