An external pressure P is applied on a cube at 0oC so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and \alpha is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:

Heat and Thermodynamics PYQ — JEE Main 2017 · Shift 0 · Q58 with Solution

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Correct answer: A

When a cube is subjected to equal pressure from all sides, it undergoes bulk compression.

By definition of bulk modulus,

K = -\frac{P}{\Delta V/V}

So, for small compression,

\frac{\Delta V}{V} = -\frac{P}{K}

Thus the volume decreases by fraction

\left|\frac{\Delta V}{V}\right| = \frac{P}{K}

For small deformations of a cube, if each side changes by fractional amount $\epsilon$ , then

\frac{\Delta V}{V} = 3\epsilon

So,

3\epsilon = -\frac{P}{K}

Hence,

\epsilon = -\frac{P}{3K}

This means each side is shortened by fractional amount

\frac{\Delta l}{l} = -\frac{P}{3K}

If the cube is heated by temperature rise $\Delta T$ , the fractional increase in length is

\frac{\Delta l}{l} = \alpha \Delta T

To bring the cube back to its original side length, thermal expansion must exactly compensate the fractional shortening:

\alpha \Delta T = \frac{P}{3K}

Therefore,

\Delta T = \frac{P}{3\alpha K}

\boxed{\Delta T = \frac{P}{3\alpha K}}

This corresponds to Option A.