Animated Solution for Physics - Properties of Solids and Liquids: A substance of mass $M\text{ kg}$ requires a power input of $P\text{ watts}$ to remain in the molten state at its melting point. When the power source is turned off, the sample completely solidifies in time $t\text{ seconds}$. The latent heat of fusion of the substance is …… .

When you see a molten substance sitting perfectly at its melting point, it might look like nothing is happening. But in reality, a fierce battle is raging at the microscopic level. The substance is constantly bleeding heat into the cooler surroundings. To prevent it from freezing, we must supply power $P$. This is the essence of a steady state: the power input perfectly balances the rate of heat loss.

Therefore, we can confidently say that the rate at which the substance loses heat to the environment is exactly $P$ Joules per second.

What happens when we flip the switch and turn the heater off? The substance doesn't instantly freeze. It is still at its melting point, meaning the temperature difference between the substance and the room is exactly the same as before. Because Newton's Law of Cooling depends on this temperature difference, the rate of heat loss remains locked at $P$.

Over the next $t$ seconds, the substance bleeds heat into the room until it completely solidifies. The total heat $Q$ lost during this time is simply the rate of heat loss multiplied by the time:

This lost heat isn't just disappearing; it is the latent heat of fusion being released as the molecules lock into a rigid solid structure. We know from calorimetry that the total latent heat released during a phase change is given by:

where $M$ is the mass and $L$ is the specific latent heat of fusion.

Equating the two expressions for the total heat lost, we get:

Solving for the specific latent heat of fusion $L$, we arrive at our final elegant result:

This problem is a beautiful reminder that in physics, "maintaining a state" often requires continuous energy, and understanding where that energy goes is the key to unlocking the system's secrets.