Answer
a) $300\;\rm nm$
b) $600\;\rm nm$
Work Step by Step
a) We know that the mean free path is given by
$$\lambda=\dfrac{V}{4\sqrt{2}\pi Nr^2}$$
So it is obvious that the mean free path is directly proportional to the volume. And since the volume is constant, the mean free path remains constant.
Thus,
$$\lambda_2=\lambda_1=\color{red}{\bf 300}\;\rm nm$$
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b) when heat is doubled at constant pressure, the final volume is then
$$\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}$$
$T_2=2T_1$,
$$\dfrac{V_1}{ \color{red}{\bf\not} T_1}=\dfrac{V_2}{2 \color{red}{\bf\not} T_1}$$
Thus,
$$V_2=2V_1$$
And since $\lambda \propto V$, and $V$ is doubled, then
$$\lambda_2=2\lambda_1=2(300)=\color{red}{\bf 600}\;\rm nm$$
