Question

An enclosed ideal gas a has its pressure p as a function of its volume v as $P={{P}_{0}}-\alpha {{V}^{2}}$, where $\alpha$ is a constant. Find the physical dimension of $\alpha$.

Answer 1

Let us find the dimensional formula of the terms in the left-hand side. As the mathematical sign used is subtraction, the dimensional formulas of both the terms on the right-hand side must be the same and they must be equal to the dimensional formula of the left-hand side.

Complete answer:
Let us find the dimensional formula of the physical quantity on the left-hand side,
Pressure has a dimensional formula as $M{{L}^{-1}}{{T}^{-2}}$.
On the right-hand side, the mathematical sign is subtraction which means the dimensional formula of both the physical quantities is the same. As the quantity on the right-hand side must have the same dimensional formula as that of the left-hand side,
We can write,
$\alpha {{V}^{2}}=M{{L}^{-1}}{{T}^{-2}}$
The dimensional formula of volume is ${{L}^{3}}$.
Therefore, we can now find the dimensional formula of unknown quantity as,
$\begin{aligned} & \alpha ({{L}^{3}})=M{{L}^{-1}}{{T}^{-2}} \\\ & \alpha =M{{L}^{-4}}{{T}^{-2}} \\\ \end{aligned}$
Hence, we can find the dimensional formula of the unknown physical quantity in this way.

Additional information:
Dimensions of a physical quantity are the power storage the fundamental units raised to obtain one unit of the quantity. Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of physical quantities. Bees dimensions are independent of the numerical multiples and constants and all the quantities in the world can be expressed as a function of the fundamental dimensions. The expression showing the power storage fundamental units are to be raised to obtain one unit of it that repoints the dimensional formula. The physical quantities which have dimensions and have a fixed value are called dimensional constant.

Note:
If the mathematical sign between the physical quantity is multiplication or division, the mode of calculating the dimensional formula will be different. In such cases, we have to multiply or divide the dimensional quantities before equating it to the left-hand side value.