$\lambda = 1/(\sqrt{2}\pi d^2 n)$; $n_B = 2n_A$, $d_B = 2d_A = 2d$
$\lambda_B = 1/(\sqrt{2}\pi(2d)^2(2n_A)) = 1/(8\sqrt{2}\pi d^2 n_A)$
$\lambda_A = 1/(\sqrt{2}\pi d^2 n_A)$
$\lambda_A/\lambda_B = 8$
Answer: 8
Assumptions: gas molecules are point masses, no intermolecular forces except during collision, collisions are perfectly elastic, molecules are in random motion. Pressure from kinetic theory: $P = \frac{1}{3}mn\overline{v^2}$. Connects microscopic (molecular motion) to macroscopic (P, T, V).
Boyle\'s law: $PV = $ const at constant $T$ (constant $N\overline{v^2}/3$). Charles\' law: $V \propto T$ at const $P$ ($\overline{v^2} \propto T$). Avogadro\'s law: equal volumes of gases at same $T, P$ contain equal molecules (same $n$, same $\overline{v^2}$ → same $P$). Dalton\'s partial pressures: $P_{total} = \sum P_i$ (no interaction).
Mean free path: $\lambda = 1/(\sqrt{2}\pi d^2 n)$. Collision frequency: $f = \sqrt{2}\pi d^2 n\overline{v}$. Mean time between collisions: $\tau = \lambda/\overline{v} = 1/(\sqrt{2}\pi d^2 n\overline{v})$. Thermal conductivity $K \propto \lambda v_{avg} C_v \rho$. Viscosity $\eta \propto \lambda v_{avg} \rho$. Diffusion coefficient $D \propto \lambda v_{avg}$. All decrease with increasing pressure (shorter $\lambda$) and increase with temperature.
Each degree of freedom contributes $\frac{1}{2}k_BT$ to average energy. Monoatomic gas: $f = 3$ (translational), $E = \frac{3}{2}k_BT$, $C_v = \frac{3}{2}R$, $\gamma = 5/3$. Diatomic (room temp): $f = 5$ (3 trans + 2 rot), $E = \frac{5}{2}k_BT$, $C_v = \frac{5}{2}R$, $\gamma = 7/5$. Diatomic (high temp): $f = 7$ (+ 2 vibrational), $C_v = \frac{7}{2}R$, $\gamma = 9/7$.
Kinetic theory defines three speeds and questions frequently ask you to distinguish them. The root mean square speed is $v_{rms} = \sqrt{3RT/M}$, the average speed is $\sqrt{8RT/\pi M}$, and the most probable speed is $\sqrt{2RT/M}$. Their ratio is fixed at $v_{rms} : v_{avg} : v_{mp} = 1.73 : 1.60 : 1.41$, so $v_{rms}$ is always the largest and $v_{mp}$ the smallest. Only $v_{rms}$ relates directly to kinetic energy.
Each degree of freedom contributes $\frac{1}{2}kT$ of energy per molecule. A monatomic gas has 3 translational degrees, giving $C_v = \frac{3}{2}R$ and $\gamma = 1.67$. A diatomic gas adds 2 rotational degrees, giving $C_v = \frac{5}{2}R$ and $\gamma = 1.40$. At high temperature vibrational modes activate and add 2 more, raising $C_v$ to $\frac{7}{2}R$. In every case $C_p - C_v = R$, which is Mayer's relation and holds for all ideal gases.
The average distance a molecule travels between collisions is $\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$, where d is the molecular diameter and n the number density. Because n falls as pressure falls, the mean free path grows in a vacuum — which is why a vacuum flask insulates so well, and why the mean free path inside a cathode ray tube is long enough for electrons to cross it without colliding.
Using grams per mole in the speed formula. M must be in kg/mol. Using 32 instead of 0.032 for oxygen changes the answer by a factor of about 32.
Thinking temperature measures speed. It measures average kinetic energy. At the same temperature a heavier molecule moves more slowly, which is exactly why the speeds differ between gases.