Reading an S-T graph of a transition

Problem

The graph shows the entropy of one mole of a substance as a function of temperature $T$, from $50\;\text{K}$ to $500\;\text{K}$. Two vertical jumps can be seen (at the melting temperature $T_f$ and at the vaporisation temperature $T_v$) and three continuous, increasing segments between them. Explain what the vertical jumps represent and what the continuous segments represent.

Solution

The vertical JUMPS are the phase transitions: melting at $T_f$ and vaporisation at $T_v$. During a transition the substance absorbs the latent heat $L$ at constant temperature, so $T$ does not change but the entropy does, with a net jump $\Delta S=\frac{L}{T}$ (that is why the segments are vertical: $\Delta T=0$, $\Delta S>0$). The vaporisation jump is larger because $L_v>L_f$.

The CONTINUOUS, increasing segments are the heating of a single phase (solid, then liquid, then gas): here the temperature rises and the entropy grows gradually according to $dS=\frac{c\,dT}{T}\ \Rightarrow\ \Delta S=\int \frac{c\,dT}{T}$ a concave growth (the slope $1/T$ decreases as $T$ increases). No latent heat here: only sensible heat. $\ev{\text{jumps}=L/T,\quad \text{segments}=\textstyle\int c\,dT/T}$

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Topics: Entropia e secondo principio Concepts: Entropia · Calore latente e cambiamenti di stato · Trasformazioni termodinamiche Skills: Lettura dei grafici