Question

The dimensional formula for $b$ is same as that for:

• A. $P$
• B. $V$
• C. $P{V}^2$
• D. $RT$

Answer 1

The correct option is B $V$

Van der Waals’ equation of state for a real gas

This equation can be derived by considering a real gas and 'converting ' it to an ideal gas.

Volume correction:

We know that for an ideal gas $P´V=nRT$. Now in a real gas the molecular volume cannot be ignored and therefore let us assume that '$b$' is the volume excluded (out of the volume of container) for the moving gas molecules per mole of a gas. Therefore due to n moles of a gas the volume excluded would be $nb.$ A real gas in a container of volume $V$ has only available volume of $(V-nb)$ and this can be thought of as an ideal gas in container of volume $(V-nb)$.

Hence, Ideal volume

$V_i=V–nb$

Pressure correction:

Let us assume that the real gas exerts a pressure $P$. The molecules that exert the force on the container will get attracted by molecules of the immediate layer which are assumed not to be exerting pressure.

It can be seen that the pressure the real gas exerts would be less than the pressure an ideal gas would have exerted. The real gas experiences attractions by its molecules in the reverse direction. Therefore if a real gas exerts a pressure $P$, then an ideal gas would exert a pressure equal to $P+p$ ($p$ is the pressure lost by the gas molecules due to attractions).

This small pressure $p$ would be directly proportional to the extent of attraction between the molecules which are hitting the container wall and the molecules which are attracting these.

Therefore $p\propto \frac{n}{v}$ (concentration of molecules which are hitting the container's wall)

$P\propto \frac{n}{v}$ (concentration of molecules which are attracting these molecules) $\to p\propto \frac{n^2}{v^2}$

$P=\frac{a{n}^2}{v^2}$ where a is the constant of proportionality which depends on the nature of gas.

A higher value of '$a$' reflects the increased attraction between gas molecules.

Hence ideal pressure

$P_i=P+\frac{a{n}^2}{v^2}$…...........(ii)

Here, $n$ = number of moles of real gas; $V$ = volume of the gas; $a$ = A constant whose value depends upon the nature of the gas

Substituting the values of ideal volume and ideal pressure in ideal gas equation $i.e.PV=nRT$, the modified equation is obtained as

$\left(\begin{array}{c}P+\frac{a}{V^2}\end{array}\right)(V-b)=RT$,

where $P$ is the pressure, $V$ is molar volume and $T$ is the temperature of the given sample of gas. $R$ is called molar gas constant, $a$ and $b$ are called van der Waals constants.

Here $b$ will the have same dimensions as $V$.

$\left[b\right]=\left[V\right]$