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(C) The van der Waals constant $b$ represents the excluded volume per mole of gas molecules.For a spherical molecule of radius $r$,the volume of one m

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$b = 4 \left[ \frac{4}{3}\pi r^3 \right] N_A$

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(C) The van der Waals constant $b$ represents the excluded volume per mole of gas molecules.For a spherical molecule of radius $r$,the volume of one molecule is $V = \frac{4}{3}\pi r^3$.The excluded volume for a pair of molecules is the volume of a sphere with radius $2r$,which is $\frac{4}{3}\pi (2r)^3 = 8 	imes \frac{4}{3}\pi r^3$.The excluded volume per molecule is half of this,i.e.,$4 	imes \frac{4}{3}\pi r^3$.Therefore,for $1 \ 	ext{mole}$ of gas ($N_A$ molecules),the constant $b$ is given by $b = 4 	imes \left( \frac{4}{3}\pi r^3 ight) N_A$.

Which of the following relations between the van der Waals constant $b$ and the radius of a spherical molecule $r$ is correct?

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