Question

The dimensional formula of compressibility is?
(A) $\left[M^0{L}^{-1}{T}^{-1}\right]$
(B) $\left[M^{-1}L{T}^2\right]$
(C) $\left[ML{T}^{-2}\right]$
(D) None of these

Answer 1

Hint To find the answer to the question, you first and foremost need to know what compressibility is and how we define it. Compressibility is the reciprocal of bulk modulus, so its dimensional formula will be the exact reciprocal of the dimensional formula of bulk modulus, which can be easily found out.

Complete step by step answer
As explained in the hint section of the solution, the easiest approach to reach at the answer is to define the dimensional formula of bulk modulus and find the reciprocal of it since compressibility is nothing but the reciprocal of bulk modulus, mathematically, we can write it as:
$K={\displaystyle \frac{1}{B}}$
Where, $K$ is the compressibility and,
$B$ is the bulk modulus of the substance.
Bulk modulus can be easily defined as the ratio of pressure and the relative change in the volume thus produced due to the pressure applied.
Mathematically, we can write it as:
$B={\displaystyle \frac{P}{\left({\displaystyle \frac{\mathrm{\Delta }V}{V}}\right)}}$
Where, $B$ is the bulk modulus of the substance,
$P$ is the pressure applied on the liquid body or particles,
$V$ is the volume of the liquid and
$\mathrm{\Delta }V$ is the change in volume brought upon due to the applied pressure $P$
Since the denominator of the term on the right-hand side of the equation is just a ratio of same quantities, the dimensional formula of bulk modulus is exactly the same as the dimensional formula of pressure, and has the same unit as that of the pressure.
Hence, we can safely say that the dimensional formula of bulk modulus is exactly the same as that of pressure and thus can be given as:
$\left[M^1{L}^{-1}{T}^{-2}\right]$
Now, since compressibility is the reciprocal of bulk modulus, its dimensional formula can be given as:
$$\begin{gathered} {\left[M^1{L}^{-1}{T}^{-2}\right]}^{-1} \\ \Rightarrow \left[M^{-1}{L}^1{T}^2\right] \end{gathered}$$

As we can see, this matches with the option (B), the correct answer to the question is option (B).

Note Many students get really confused between compressibility and bulk modulus and thus tick the wrong option as the answer. The concept is simple, bulk modulus has the same unit and dimensions as that of pressure, while compressibility is the reciprocal of bulk modulus and thus has the unit and dimensional formula as that of reciprocal of bulk modulus.