Bulk modulus and compressibility — why are liquids nearly incompressible

easyCBSE-11JEE-MAIN•NCERT Class 11•3 min read
TagsKinematics

Question

A liquid has a bulk modulus of 2.2×1092.2 \times 10^9 Pa. If a pressure of 1.1×1071.1 \times 10^7 Pa is applied to 1 litre of this liquid, find the decrease in volume. Also explain why liquids are considered nearly incompressible compared to gases.

(NCERT Class 11, Mechanical Properties of Fluids)

Solution — Step by Step

Recall the bulk modulus formula

Bulk modulus BB measures a material's resistance to uniform compression. It is defined as:

B=−ΔPΔV/VB = -\frac{\Delta P}{\Delta V / V}

The negative sign indicates that volume decreases when pressure increases.

Rearrange to find volume change

We need ΔV\Delta V:

ΔV=−ΔP×VB\Delta V = -\frac{\Delta P \times V}{B}

Substituting: ΔP=1.1×107\Delta P = 1.1 \times 10^7 Pa, V=1V = 1 L =10−3= 10^{-3} m3^3, B=2.2×109B = 2.2 \times 10^9 Pa.

Calculate the decrease in volume

ΔV=−1.1×107×10−32.2×109=−1.1×1042.2×109=−5×10−6 m3\Delta V = -\frac{1.1 \times 10^7 \times 10^{-3}}{2.2 \times 10^9} = -\frac{1.1 \times 10^4}{2.2 \times 10^9} = -5 \times 10^{-6} \text{ m}^3

That's just 0.005 mL out of 1000 mL. The fractional change is only 0.5%. This is why we call liquids nearly incompressible.

Compare with gases

For an ideal gas at atmospheric pressure, the bulk modulus equals the pressure itself — roughly 10510^5 Pa. Compare that with the liquid's 2.2×1092.2 \times 10^9 Pa. The liquid's bulk modulus is about 22,000 times larger than the gas. Same applied pressure would compress a gas dramatically but barely affects a liquid.

Why This Works

At the molecular level, liquid molecules are already closely packed — there's very little empty space between them. When you apply pressure, you're essentially trying to push molecules that are already touching even closer. The intermolecular repulsive forces resist this strongly.

Gases, on the other hand, have huge gaps between molecules. Pressure simply pushes those far-apart molecules closer together, which is easy. That's why compressibility (κ=1/B)(\kappa = 1/B) of gases is orders of magnitude higher than liquids.

Solids have even higher bulk moduli than liquids because their molecules are locked in a rigid lattice. The order is: Bsolid>Bliquid≫BgasB_{\text{solid}} > B_{\text{liquid}} \gg B_{\text{gas}}.

Alternative Method

You can also think in terms of compressibility κ=1/B\kappa = 1/B:

κ=12.2×109≈4.5×10−10 Pa−1\kappa = \frac{1}{2.2 \times 10^9} \approx 4.5 \times 10^{-10} \text{ Pa}^{-1}

Then ΔV/V=−κ⋅ΔP=−4.5×10−10×1.1×107=−4.95×10−3≈−0.5%\Delta V / V = -\kappa \cdot \Delta P = -4.5 \times 10^{-10} \times 1.1 \times 10^7 = -4.95 \times 10^{-3} \approx -0.5\%.

💡 Expert Tip

In CBSE board exams, they often ask "why are liquids nearly incompressible?" as a 2-mark theory question. The answer is two lines: high bulk modulus (due to closely packed molecules) and very small fractional volume change even under large pressures.

Common Mistake

⚠️ Common Mistake

Students sometimes forget the negative sign in the bulk modulus formula and report a positive ΔV\Delta V, implying the volume increased under compression. Remember: BB is defined as a positive quantity, so the formula already accounts for the sign. When you solve for ΔV\Delta V, the result should be negative (volume decreases). If you get a positive answer, something went wrong with your signs.

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