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Properties of Solids and Liquids: Physics | JEE Main

Under isothermal condition, the pressure of a gas is given by

\text{P} = \text{a}\text{V}^{-3}

, where

\text{a}

is a constant and

\text{V}

is the volume of the gas. The bulk modulus at constant temperature is equal to

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\frac{\mathrm{P}}{2}

3\text{ P}

2\text{ P}

\text{P}

Step-by-step solutionView

Correct answer

Using the definition of bulk modulus $B = -V \frac{\mathrm{d}P}{\mathrm{d}V}$ on $P = aV^{-3}$ , we find $B = -V(-3aV^{-4}) = 3aV^{-3} = 3P$ .

Option analysis

Why each option works or fails

A ·

\frac{\mathrm{P}}{2}

Believing that the bulk modulus is divided by the power or index rather than multiplied by it. Differentiate $P = aV^{-3}$ with respect to $V$ to get $\frac{\mathrm{d}P}{\mathrm{d}V} = -3aV^{-4}$ , then multiply by $-V$ to obtain $3P$ .

B ·

3\text{ P}

This is the correct answer. Evaluating $B = -V \frac{\mathrm{d}P}{\mathrm{d}V} = -V(-3aV^{-4}) = 3(aV^{-3}) = 3P$ directly gives $3P$ .

C ·

2\text{ P}

Incorrectly applying the power rule as decreasing the magnitude of the exponent from 3 to 2. The derivative of $V^{-3}$ brings down a factor of $-3$ , giving a multiplier of $3$ after accounting for the negative sign in the bulk modulus formula.

D ·

\text{P}

Assuming the isothermal bulk modulus of any gas is always equal to $P$ , which is only true for an ideal gas where $PV = \text{constant}$ . Use the given equation of state $P = aV^{-3}$ ( $PV^3 = a$ ) rather than the standard ideal gas relation $PV = \text{constant}$ .

Reviewed route

Solution

StepWorking

01given

Pressure-volume relation: $P = aV^{-3}$ where $a$ is a constant.

02find

The bulk modulus $B$ of the gas at constant temperature.

03strategise

Bulk modulus is defined as $B = -V \frac{dP}{dV}$ . Differentiate $P = aV^{-3}$ with respect to $V$ and substitute into the definition.

04execute

Differentiating $P$ with respect to $V$ gives $\frac{dP}{dV} = -3aV^{-4} = -3\frac{P}{V}$ . Then, $B = -V \left(-3\frac{P}{V}\right) = 3P$ .

✓verify

For a polytropic process $PV^n = \text{const}$ , the bulk modulus is $nP$ . Here $n = 3$ , so $B = 3P$ , which is positive and physically consistent.

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Physics
Concepts assessed: Physics
Academic status: Reviewed by official_key
Source: pyq
Editorial review: 8 September 2026

Quick checks

Students also ask

Why is there a negative sign in the bulk modulus definition?

Because an increase in pressure produces a decrease in volume, so $dV/dP$ is negative. The negative sign ensures that the bulk modulus is a positive physical quantity.

Answer

Using the definition of bulk modulus $B = -V \frac{\mathrm{d}P}{\mathrm{d}V}$ on $P = aV^{-3}$ , we find $B = -V(-3aV^{-4}) = 3aV^{-3} = 3P$ .

Why each option works or fails

A: $\frac{\mathrm{P}}{2}$ - Believing that the bulk modulus is divided by the power or index rather than multiplied by it. Differentiate $P = aV^{-3}$ with respect to $V$ to get $\frac{\mathrm{d}P}{\mathrm{d}V} = -3aV^{-4}$ , then multiply by $-V$ to obtain $3P$ .
B · correct: $3\text{ P}$ - This is the correct answer. Evaluating $B = -V \frac{\mathrm{d}P}{\mathrm{d}V} = -V(-3aV^{-4}) = 3(aV^{-3}) = 3P$ directly gives $3P$ .
C: $2\text{ P}$ - Incorrectly applying the power rule as decreasing the magnitude of the exponent from 3 to 2. The derivative of $V^{-3}$ brings down a factor of $-3$ , giving a multiplier of $3$ after accounting for the negative sign in the bulk modulus formula.
D: $\text{P}$ - Assuming the isothermal bulk modulus of any gas is always equal to $P$ , which is only true for an ideal gas where $PV = \text{constant}$ . Use the given equation of state $P = aV^{-3}$ ( $PV^3 = a$ ) rather than the standard ideal gas relation $PV = \text{constant}$ .

Step-by-step solution

given: Pressure-volume relation: $P = aV^{-3}$ where $a$ is a constant.
find: The bulk modulus $B$ of the gas at constant temperature.
strategise: Bulk modulus is defined as $B = -V \frac{dP}{dV}$ . Differentiate $P = aV^{-3}$ with respect to $\text{P}$ and substitute into the definition.
execute: Differentiating $P$ with respect to $V$ gives $\frac{dP}{dV} = -3aV^{-4} = -3\frac{P}{V}$ . Then, $B = -V \left(-3\frac{P}{V}\right) = 3P$ .
verify: For a polytropic process $PV^n = \text{const}$ , the bulk modulus is $nP$ . Here $n = 3$ , so $B = 3P$ , which is positive and physically consistent.

Shortcut:

When to use it: For any process of the form PV^n = constant

quick_kill: Rewrite $P = aV^{-3}$ as $PV^3 = a$ . For any process $PV^n = \text{constant}$ , the bulk modulus is $B = nP$ . Since $n=3$ , $B = 3P$ .

Properties of Solids and Liquids: Physics | JEE Main

What feels right?

Using the definition of bulk modulus $B = -V \frac{\mathrm{d}P}{\mathrm{d}V}$ on $P = aV^{-3}$ , we find $B = -V(-3aV^{-4}) = 3aV^{-3} = 3P$ .

Why each option works or fails

Solution

01given

02find

03strategise

04execute

✓verify

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Properties of Solids and Liquids: Physics | JEE Main

Using the definition of bulk modulus B=−VdPdVB = -V \frac{\mathrm{d}P}{\mathrm{d}V}B=−VdVdP​ on P=aV−3P = aV^{-3}P=aV−3, we find B=−V(−3aV−4)=3aV−3=3PB = -V(-3aV^{-4}) = 3aV^{-3} = 3PB=−V(−3aV−4)=3aV−3=3P.

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