Bulk Modulus and Volume Change Calculation

❓ Question

A sample of a liquid is kept at:

$$1\text{ atm}$$

It is compressed to:

$$5\text{ atm}$$

which leads to change of volume of:

$$0.8\text{ cm}^3$$

If the bulk modulus of the liquid is:

$$2\text{ GPa}$$

the initial volume of the liquid was ______ litre.

(Take 1 atm = 10⁵ Pa)

🖼 Question Image

✍️ Short Explanation

This problem is based on:

👉 Bulk modulus
👉 Compressibility of liquids
👉 Pressure-volume relation

Main idea:

$$B=\frac{\Delta P\cdot V}{\Delta V}$$

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🔷 Step 1 — Write Given Values 💯

Initial pressure:

$$P_1=1\text{ atm}$$

Final pressure:

$$P_2=5\text{ atm}$$

Pressure change:

$$\Delta P=5-1$$
$$=4\text{ atm}$$

Convert into SI units:

$$\Delta P=4\times10^5\text{ Pa}$$

Bulk modulus:

$$B=2\text{ GPa}$$
$$=2\times10^9\text{ Pa}$$

Volume change:

$$\Delta V=0.8\text{ cm}^3$$

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🔷 Step 2 — Apply Bulk Modulus Formula

Using:

$$B=\frac{\Delta P\cdot V}{\Delta V}$$

Rearranging:

$$V=\frac{B\Delta V}{\Delta P}$$

Substitute values:

$$V= \frac{ (2\times10^9)(0.8) }{ 4\times10^5 }$$

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🔷 Step 3 — Simplify

$$V= \frac{1.6\times10^9}{4\times10^5}$$
$$= 0.4\times10^4$$
$$=4000\text{ cm}^3$$

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🔷 Step 4 — Convert into Litres

Since:

$$1000\text{ cm}^3=1\text{ litre}$$
$$V=\frac{4000}{1000}$$
$$V=4\text{ litre}$$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Pressure difference calculate karna bhool jaana

❌ GPa ko Pa me convert na karna

❌ cm$^3$ ko litre me convert galat kar dena

Remember:

$$\boxed{ B=\frac{\Delta P\cdot V}{\Delta V} }$$

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✅ Final Answer

$$\boxed{4\text{ litre}}$$

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📚 Related Topics

• Extension of Wire: Length vs Thickness
• Ratio of Elongation in Two Wires
• Young Modulus Calculation from Load and Extension