Which of the following expression represents the CORRECT relationship between the compressibility (C) and bulk modulus (K) of the fluid?
Try it first: pick an option, then read the explanation.
Correct answer: C. C = 1 / K
Asked in SSC JE 25-01-2018 (Evening Shift)
Detailed Explanation
Bulk modulus (K) describes a fluid's stiffness and resistance to volumetric compression. Compressibility (C or β) quantifies how readily a fluid contracts under pressure.
\[ K = -\frac{dp}{dV/V} = \rho \frac{dp}{d\rho} \]
\[ C = \frac{1}{K} = -\frac{dV/V}{dp} \]
Why “C = 1 / K” Is Correct
By definition, compressibility is the exact mathematical inverse of bulk modulus: C = 1/K.
| Property | Formula | SI unit | Standard value (water, 4°C) |
|---|---|---|---|
| Density ρ | m/V | kg/m³ | 1000 kg/m³ |
| Specific weight γ | ρg | N/m³ | 9810 N/m³ |
| Specific gravity S | ρ/ρ_water | — | 1.0 |
| Specific volume v | 1/ρ | m³/kg | 0.001 m³/kg |
| Bulk modulus K | −dp/(dV/V) | Pa | 2.06 GPa |
| Compressibility β | 1/K | Pa⁻¹ | 4.85 × 10⁻¹⁰ Pa⁻¹ |
Why the Other Options Are Wrong
- A. C = 1 / √K: Compressibility is linear with 1/K, not the square root.
- B. C = √K: Compressibility decreases as bulk modulus increases, so C cannot equal √K.
- D. C = K^(3/2): Power 3/2 has no theoretical basis in continuum mechanics.
Key Point
Compressibility C = 1 / K; higher bulk modulus means lower compressibility.
Key Concepts for Students
- Incompressible fluids have K → ∞ and C → 0.
- Liquids have very high K (~2 GPa for water), making them nearly incompressible.
- Gases have low K (~0.1 MPa), making them highly compressible.
Practice more: this is question 38 in Properties of Fluid MCQs – Part 3. Try the whole set and check every answer instantly.