Question

Assertion. Extended volume or co-volume equals to (\[V - nb\] ) for \[n\] moles.
Reason. Co-volume depends on the effective size of gas molecules.
A. Both assertion and reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect.

Answer 1

Hint:The extended volume is the increase in volume of a gas molecule in a void space. The number of gas molecules has a definite effect on the co-volume or extended volume.

Complete step by step answer:The gas equation of n moles of the ideal gas is given as,
\[PV = nRT\], where
\[P\] is the pressure, \[V\] is the volume, \[n\] is the moles of gas, \[R\] is gas constant and \[T\] is the temperature.
Now let us assume that all gas molecules are hard spheres having the same finite radius \[r\] which is the Van Der Waals radius. In order to keep the volume fixed the available void space decreases as the size or the radius of the particle increases. For \[n\] moles of the gas, the volume must be replaced with\[V - nb\] , where \[b\] is called the excluded volume or co-volume. The gas equation is modified as:
$P = \dfrac{{RT}}{{V - nb}}$
We now have to determine the dependence of gas extended volume on the size of gas molecules. For this, consider that a particle is surrounded by a sphere of radius \[2r\] (extended) which is twice the original radius. As the distance between the centers of two gas molecules is smaller than \[2r\], thus the two particles must penetrate each other. But as the molecules of gas molecules are presumed to be hard spheres, hence such type of penetration does not take place.

Therefore the excluded volume for the two particles with average diameter \[d\] or radius \[r\] is:
$\dfrac{b}{2} = \dfrac{{4\Pi {d^3}}}{3} = 8\dfrac{{4\Pi {r^3}}}{3}$
When two particles are colliding the extended volume per particle has to be divided by two,
\[b' = \dfrac{{\dfrac{b}{2}}}{2} = \dfrac{{8\dfrac{{4\Pi {r^3}}}{3}}}{2} = 4\left[ {\dfrac{{4\Pi {r^3}}}{4}} \right]\]
Hence the co-volume or excluded volume of the molecules in motion is four times the actual volume or size of the molecules.
Thus option A is the correct answer, i.e. both assertion and reason are correct and reason is the correct explanation for assertion.

Note:As the extended volume \[b'\] is four times the actual volume of the particle. It has to be kept in mind that the extended volume is directly proportional to the diameter of the particle. The change in volume i.e. \[V - nb\] is referred to as the correction factor of volume.