Question

In the equation \[P=\dfrac{RT}{V-b}{{e}^{\dfrac{aV}{RT}}}\]
\[V=volume\], \[P=pressure\], \[R=\text{universal gas constant}\], and \[\text{T= temperature}\]. The dimensional formula of a is same as that of
A. V
B. P
C. T
D. R

Answer 1

Hint: The given relation is similar to the ideal gas equation. So, we can compare it with the ideal gas equation. This is the simplest way to find the dimensional formula of a physical quantity, if we don’t know all the components in the equation. The dimensional formula will be the same, if they are similar in physical nature.

Complete step-by-step answer:
This equation is similar to the ideal gas equation.
According to the ideal gas equation,
\[PV=nRT\], where \[V\] is the volume, \[P\] is the pressure, \[T\] is the temperature, \[n\] is the number of moles and \[R\] is the universal gas constant.
The given equation can be written as,
\[P(V-b)=RT{{e}^{\dfrac{aV}{RT}}}\]
Therefore, \[^{\dfrac{aV}{RT}}\] is a dimensionless quantity.
So, we can find the dimension of \[a\].
\[a\propto \dfrac{RT}{V}\]
From the ideal gas equation, we can conclude that,
\[P\propto \dfrac{RT}{V}\], \[n\] is the number and doesn't have any dimensional equation.
It represents \[a\] is dimensionally similar to the pressure.
So, the correct option is B.

Additional information:
Ideal gas equation is actually the sum of several gas laws. It includes Boyle's law, Charles law and Avogadro law. Each of the gas laws contributed each term in the ideal gas equation.
From Boyle’s law; \[V\propto \dfrac{1}{P}\] (at constant n and T)
From Charles law; \[V\propto T\] (at constant n and P)
From Avogadro’s law; \[V\propto n\] (at constant P and T)
Collectively we can write as,
\[V\propto \dfrac{nT}{P}\]
Finally, we added a universal gas constant as a proportionality constant in this equation.
\[PV=nRT\]
Therefore, an ideal gas can be stated as the gas which obeys above mentioned gas laws.
Now we can discuss the key concepts of kinetic theory of gases.
i. Gases consist of small particles which are in continuous and random motion.
ii. The volume of the molecules is negligible compared to the total volume occupied by the gas.
iii. Intermolecular forces are negligible.
iv. Pressure is due to the gas molecules colliding with the walls of the container.

Note: By doing comparison we will get, \[^{\dfrac{aV}{RT}}\] as a dimensionless quantity. But don’t stop at this point. Even if it is collectively dimensionless, individual terms have its own dimensional formula. That maybe gets cancelled during the calculation. That’s why we are getting dimensions of unknown quantities similar to the pressure.