Question

The variation of volume of liquid with respect to the pressure and its reciprocal respectively are

• A. Compressibility & Young's modulus
• B. Young's modulus & Compressibility
• C. Compressibility & Bulk modulus
• D. Bulk modulus & Compressibility

Hint:

Remember the physical analogy to solid mechanics: - Bulk Modulus ($K$) measures volumetric stiffness (high $K$ means highly incompressible, like water). - Compressibility ($\beta$) measures volumetric compliance. - They are related by an inverse relationship: $\beta = \frac{1}{K}$.

Answer 1

The Correct Option is C

Concept: To understand the volumetric behavior of a fluid under changing external pressures, fluid mechanics introduces two foundational parameters: the Bulk Modulus of Elasticity (\(K\)) and Compressibility (\(\beta\)). Let us analyze a specified mass of fluid that initially occupies an equilibrium volume \(V\) under an ambient static pressure \(P\). When an incremental external pressure increase \(dP\) is applied uniformly across the boundaries of this fluid, the fluid compresses, causing its volume to decrease by an amount \(dV\). The corresponding volumetric strain (\(\epsilon_v\)) is given by: \[ \epsilon_v = \frac{\text{Change in Volume}}{\text{Original Volume}} = \frac{-dV}{V} \] The negative sign indicates a physical reduction in volume as pressure increases.

Step 1: Defining the Bulk Modulus (\(K\)).
The Bulk Modulus describes the fluid's resistance to volumetric compression. It is defined as the ratio of the incremental pressure change to the resulting volumetric strain: \[ K = \frac{\text{Direct Incremental Pressure}}{\text{Volumetric Strain}} = \frac{dP}{\left(-\frac{dV}{V}\right)} = -V \frac{dP}{dV} \]

Step 2: Defining Compressibility (\(\beta\)).
Compressibility represents the ease with which a fluid undergoes a volume change when subjected to a pressure gradient. Mathematically, compressibility is the fractional change in volume per unit change in pressure. It is defined as: \[ \beta = -\frac{1}{V} \frac{dV}{dP} \] Comparing the formulas for Bulk Modulus (\(K\)) and Compressibility (\(\beta\)), we see that they are exact reciprocals of each other: \[ \beta = \frac{1}{K} \]

Step 3: Matching the question criteria.
The question asks for two specific definitions in sequence:
• The variation of volume of liquid with respect to pressure: This describes fractional volume change per unit pressure change, which is Compressibility (\(\beta\)).
• Its reciprocal: The reciprocal of compressibility (\(\frac{1}{\beta}\)) equals the Bulk modulus (\(K\)). Therefore, the correct pair in order is Compressibility & Bulk modulus, matching Option (3).