Animated Solution for Physics - Properties of Solids and Liquids: Liquid oxygen at 50 K is heated to 300 K at constant pressure of 1 atm. The rate of heating is constant. Which of the following graphs represent the variation of temperature with time?

When we first learn about calorimetry, we are often taught the simple formula $dQ=mCdT$. For everyday substances like water at room temperature, it is perfectly fine to assume that the specific heat capacity $C$ is a constant. This assumption leads to a beautifully straight, linear graph when we plot temperature against time under a constant heating rate.

However, the universe behaves very differently near absolute zero. When dealing with substances like liquid oxygen at $50\text{ K}$, quantum mechanics takes the wheel. According to Debye's theory, the specific heat capacity of solids and liquids at very low temperatures is highly dependent on the temperature itself. As the temperature rises, the specific heat capacity $C$ also increases. This single fact completely changes the geometry of our heating curve.

Let's translate this physical reality into mathematics. We are given that the rate of heating is constant. Let's call this constant power $P$.

We know that:

Substituting $dQ=mCdT$ into our power equation, we get:

To understand the shape of the Temperature-Time ($T-t$) graph, we need to isolate its slope, $\frac{dT}{dt}$:

Look closely at this master equation. The power $P$ is constant, and the mass $m$ is constant. The only variable on the right side is the specific heat $C$, which sits squarely in the denominator.

As the liquid oxygen absorbs heat, its temperature $T$ increases. As we established earlier, an increasing temperature causes the specific heat $C$ to increase.

Because $C$ is growing larger and it is in the denominator, the overall fraction $\frac{P}{mC}$ must grow smaller. Therefore, the slope $\frac{dT}{dt}$ is continuously decreasing.

Geometrically, a decreasing slope means that the curve is getting flatter as time goes on. This creates a concave downward shape.

Once the liquid oxygen reaches its boiling point (around $90\text{ K}$), it undergoes a phase change. During this time, all the supplied heat goes into breaking intermolecular bonds rather than raising the temperature. This gives us a perfectly horizontal line on the graph.

After boiling, the oxygen becomes a gas and begins to heat up again. The specific heat of the gas also slightly increases (or remains roughly constant) as vibrational modes begin to activate, meaning the slope will again decrease or remain constant, maintaining the concave downward or straight trajectory.

When we evaluate the given options, we are looking for a graph that starts with a concave downward curve, flattens out into a horizontal line, and then continues with another concave downward (or straight) curve. Graph (c) is the only option that flawlessly captures this physical reality.