Answer
See the detailed answer below.
Work Step by Step
a) We know, from ideal gas law, that
$$PV=Nk_BT$$
Hence,
$$P=\dfrac{N}{V}k_BT$$
Plugging the known;
$$P=\left(\dfrac{1}{1\times 10^{-6}}\right)(1.38\times 10^{-23})(3)$$
$$P=\color{red}{\bf 4.14 \times 10^{-17}}\;\rm Pa=\color{red}{\bf 4.1\times 10^{-22}}\;\rm atm$$
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b) We know that the rms speed is given by
$$v_{\rm rms}=\sqrt{\dfrac{3k_BT}{m_H}}$$
Plugging the known;
$$v_{\rm rms}=\sqrt{\dfrac{3(1.38\times 10^{-23})(3)}{1\times 1.661\times 10^{-27}}}=\color{red}{\bf 273}\;\rm m/s$$
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c) We know that the thermal energy of monatomic hydrogen atoms at low temperatures if given by
$$E_{th}=\frac{3}{2}nRT$$
where $n$ is given by the ideal gas law of $PV=nRT$, so $n=PV/RT$
$$E_{th}=\frac{3}{2}\dfrac{PV}{ \color{red}{\bf\not} R \color{red}{\bf\not} T} \color{red}{\bf\not} R \color{red}{\bf\not} T$$
$$E_{th}=1.5 P(L^3)$$
Solving for $L$;
$$L=\sqrt[3]{\dfrac{E_{th}}{1.5P}}$$
Plugging the known;
$$L=\sqrt[3]{\dfrac{(1)}{1.5(4.14 \times 10^{-17})}}$$
$$L=\color{red}{\bf 2.52\times 10^{25}}\;\rm m$$
which is greater than the radius of our Milky Way Galaxy by more than 50,000 times!
