Entropy of a solid or liquid

For a solid or liquid body, whose volume changes negligibly, entropy changes only because of the change in temperature.

Key formula — Entropy of a solid or liquid

$\ev{\Delta S_\text{solid} = m\,c_s\,\ln\frac{T_B}{T_A}}$

Here $m$ is the mass of the body and $c_s$ its specific heat. The structure is the same as the temperature term for the gas, but with $mc_s$ (the body’s heat capacity) in place of $nC_v$.

The sign follows directly from the logarithm:

• if the body heats up ($T_B > T_A$) then $\ln(T_B/T_A) > 0$ and the entropy increases;
• if the body cools down ($T_B < T_A$) then $\ln(T_B/T_A) < 0$ and the entropy decreases.

The fact that the entropy of a body can decrease violates no principle: the second law concerns the total entropy (body plus surroundings), not that of a single part. A cooling cup of coffee sees its own entropy fall, but the surroundings that receive that heat increase their own by a greater amount.

Watch the temperature unit

In entropy formulas the temperatures $T_A$ and $T_B$ must always be expressed in kelvin: they appear in a ratio inside a logarithm, and only the absolute scale makes that ratio physically meaningful.

This formula holds as long as the body does not change state. During melting or boiling, which happen at constant temperature while absorbing the latent heat $L$, the change in entropy is instead calculated as $\Delta S = Q/T = mL/T$, with $T$ the (constant) temperature of the transition.

Links

Topics: Entropy and the second law Concepts: Entropy · Specific heat and heat capacity Skills: Entropy balance

Related exercises: Worked exercise — the coffee cools down · Hot shower: ΔS of the universe · Cooling iron: maximum work