Physics Q34 – JEE-MAIN 2026

Home Physics ThermodynamicsQuestion #512

An ideal gas at pressure

P

and temperature

T

is expanding such that

P T^{3} = constant

. The coefficient of volume expansion of the gas is _______.

Clue 1 of 3: Definition of Coefficient of Volume Expansion

The coefficient of volume expansion $β$ for a process is defined as the fractional change in volume per unit change in temperature, i.e., $β = \frac{1}{V} \frac{d V}{d T}$ .

🧠 Full Solution Path

Step 1: Relate Volume, Pressure, and Temperature✦ Active

The ideal gas law is $P V = n R T$ . From this, we can express volume $V$ as $V = \frac{n R T}{P}$ . The given process condition is $P T^{3} = K$ , where $K$ is a constant. We can express pressure $P$ as $P = \frac{K}{T^{3}}$ . Substituting this into the ideal gas law gives the relationship between $V$ and $T$ for this process:

V = \frac{n R T}{(K / T^{3})} = \frac{n R T^{4}}{K}

Step 2: Calculate the Derivative of Volume with respect to Temperature○ Expand

The coefficient of volume expansion $β$ is defined as $β = \frac{1}{V} (\frac{d V}{d T})$ . First, differentiate the expression for $V$ with respect to $T$ :

\frac{d V}{d T} = \frac{d}{d T} (\frac{n R T^{4}}{K}) = \frac{n R}{K} (4 T^{3})

💡 Teacher's Secret Hint

Remember to treat $n$ , $R$ , and $K$ as constants during differentiation.

Step 3: Determine the Coefficient of Volume Expansion○ Expand

Now, substitute the expressions for $V$ and $\frac{d V}{d T}$ into the definition of $β$ :

β = \frac{1}{V} (\frac{d V}{d T}) = \frac{1}{(\frac{n R T^{4}}{K})} (\frac{4 n R T^{3}}{K})

β = \frac{K}{n R T^{4}} \cdot \frac{4 n R T^{3}}{K} = \frac{4}{T}

Thus, the coefficient of volume expansion of the gas is $\frac{4}{T}$ .

💡 Teacher's Secret Hint

Alternatively, one could use logarithmic differentiation: $\ln V = \ln (n R) + \ln T - \ln P$ and $\ln P + 3 \ln T = \ln K$ . Differentiating both with respect to $T$ and combining them yields the same result.

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Physics Question 34 – JEE-MAIN 2026

Clue 1 of 3: Definition of Coefficient of Volume Expansion

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