Phase graph T(t)

Problem

The graph $T(t)$ shows the heating of $1\;\text{kg}$ of a substance receiving constant power $P = 1000\;\text{W}$. Identify the solid, liquid and gas phases. Estimate the specific heats of the three phases and the latent heats of fusion and vaporisation.

Solution

Reading the graph. The sloped stretches are heating periods of a single phase: the substance absorbs heat and the temperature rises. The horizontal plateaus are phase transitions: the temperature stays constant while all the absorbed heat serves as latent heat to change state.

• First sloped stretch → heating of the solid.
• First plateau → fusion (solid → liquid).
• Second sloped stretch → heating of the liquid.
• Second plateau → vaporisation (liquid → gas).
• Last sloped stretch → heating of the vapour.

Specific heats (sloped stretches). On the stretch where the substance heats by $\Delta T$ in a time $\Delta t$, the heat supplied is $P\,\Delta t$, so: $c = \frac{P\,\Delta t}{m\,\Delta T}$ with $P = 1000\;\text{W}$ and $m = 1\;\text{kg}$. For equal power, a shallower slope (more time needed for the same $\Delta T$) corresponds to a higher specific heat.

Latent heats (plateaus). On the plateau of duration $\Delta t_\text{plateau}$: $L = \frac{P\,\Delta t_\text{plateau}}{m}$ The first plateau gives the latent heat of fusion, the second that of vaporisation (usually much longer, so a greater $L$).

$\ev{\text{plateau} = \text{transitions};\ c = P\Delta t/(m\Delta T);\ L = P\Delta t/m}$

Links

Topics: Thermodynamics Concepts: Latent heat and changes of state · Specific heat and heat capacity Skills: Reading graphs Methods: Temperature-heat diagram