7.1 Entropy change of pure substances

Entropy in Thermodynamics

Definition and Significance

Entropy is a thermodynamic property that quantifies the microscopic disorder or randomness within a system. More precisely, it measures the number of microstates consistent with a given macrostate.

A few things make entropy especially important:

• Entropy is a state function, meaning it depends only on the initial and final states of the system, not on the path taken between them. This is what allows us to choose any convenient path (including a reversible one) to calculate $\Delta S$.
• It determines the direction of spontaneous processes and sets limits on the maximum useful work a system can deliver.
• The second law of thermodynamics guarantees that the entropy of an isolated system never decreases. It either increases (irreversible process) or stays the same (reversible process).
• The SI unit of entropy is joules per kelvin ($\text{J/K}$), or on a per-mass basis, $\text{J/(kg·K)}$.

Relationship with Heat and Temperature

The fundamental definition of an entropy change for a reversible process is:

$\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}$

This tells you two things at once:

• Entropy change is directly proportional to the reversible heat transfer $\delta Q_{\text{rev}}$.
• It is inversely proportional to the absolute temperature $T$ at which that transfer occurs.

So adding a given amount of heat to a cold system produces a larger entropy increase than adding the same heat to a hot system. That inverse relationship with temperature comes up constantly in entropy calculations.

Calculating Entropy Changes

Pure Substances and Various Processes

Because entropy is a state function, you can always compute $\Delta S$ by finding any reversible path between the same two end states. The formulas below are the standard results for common process types.

• General (any reversible path):

$\Delta S = \int_{1}^{2} \frac{\delta Q_{\text{rev}}}{T}$

• Isothermal process (constant $T$):

The temperature comes out of the integral, giving $\Delta S = \frac{Q}{T}$ where $Q$ is the total heat transferred and $T$ is the constant absolute temperature.

• Isobaric process (constant pressure):

Using the specific heat at constant pressure $c_p$, $\Delta S = m \, c_p \ln\!\left(\frac{T_2}{T_1}\right)$ where $T_1$ and $T_2$ are the initial and final absolute temperatures and $m$ is the mass. On a per-unit-mass basis, drop the $m$.

• Isochoric process (constant volume):

Using the specific heat at constant volume $c_v$, $\Delta S = m \, c_v \ln\!\left(\frac{T_2}{T_1}\right)$ Note that $\Delta S$ is not zero here. Heat can absolutely be transferred at constant volume (think of heating a rigid sealed tank). The entropy change is zero only for a process that is both adiabatic and reversible (isentropic).

Phase Transitions and Entropy Changes

During a phase change at constant temperature and pressure, the entropy change is:

$\Delta S = \frac{\Delta H}{T}$

where $\Delta H$ is the enthalpy (latent heat) of the transition and $T$ is the absolute temperature at which it occurs.

• Phase transitions involve a large jump in entropy because the molecules gain significantly more freedom of motion. A solid melting into a liquid, or a liquid vaporizing into a gas, both increase disorder.
• The entropy of vaporization is much larger than the entropy of fusion for a given substance. For example, water's entropy of vaporization ($\approx 109 \; \text{J/(mol·K)}$) dwarfs its entropy of fusion ($\approx 22 \; \text{J/(mol·K)}$), because the jump in molecular disorder from liquid to gas is far greater than from solid to liquid.

Putting It Together: Multi-Step Entropy Calculation

Real problems often combine several of these steps. For instance, to find the total entropy change when ice at $-10°\text{C}$ is heated to steam at $120°\text{C}$:

1. Heat the ice from $-10°\text{C}$ to $0°\text{C}$ (use $c_p$ of ice, isochoric/isobaric formula).
2. Melt the ice at $0°\text{C}$ (use $\Delta S = \Delta H_{\text{fus}}/T$).
3. Heat the liquid water from $0°\text{C}$ to $100°\text{C}$ (use $c_p$ of liquid water).
4. Vaporize the water at $100°\text{C}$ (use $\Delta S = \Delta H_{\text{vap}}/T$).
5. Heat the steam from $100°\text{C}$ to $120°\text{C}$ (use $c_p$ of steam).

Add all five $\Delta S$ values. This works because entropy is a state function, so the total change equals the sum along any path connecting the two states.

Entropy and the Second Law

Spontaneous Processes and Irreversibility

The second law of thermodynamics states that the total entropy of an isolated system can never decrease. For any real process:

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \geq 0$

• If $\Delta S_{\text{universe}} > 0$, the process is irreversible (and spontaneous in that direction).
• If $\Delta S_{\text{universe}} = 0$, the process is reversible (an idealization).
• If $\Delta S_{\text{universe}} < 0$, the process is impossible as described.

This is what gives thermodynamic processes a preferred direction. Heat flows from hot to cold, gases expand into vacuums, and mixed gases don't spontaneously unmix, all because the reverse would violate the second law.

Clausius Inequality

The Clausius inequality provides the mathematical backbone of the second law:

$\oint \frac{\delta Q}{T} \leq 0$

For a process between two states (not necessarily a cycle), this becomes:

$\Delta S \geq \int \frac{\delta Q}{T}$

• For a reversible process: $\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}$ (equality holds).
• For an irreversible process: $\Delta S > \int \frac{\delta Q}{T}$ (strict inequality).

The key takeaway: the actual entropy change of a system during an irreversible process is larger than what you'd calculate from the actual heat transfer divided by temperature. The "extra" entropy is generated internally by irreversibilities.

Reversible vs. Irreversible Processes

Characteristics of Reversible Processes

A reversible process can be reversed so that both the system and surroundings return exactly to their original states, with zero net entropy generated in the universe.

• The system passes through a continuous sequence of equilibrium states (quasi-static).
• The entropy change equals $\int \delta Q_{\text{rev}} / T$ exactly.
• Reversible processes are idealizations. They require no friction, no unrestrained expansion, no heat transfer across a finite temperature difference, and infinite time to complete.
• Classic examples: isothermal expansion/compression of an ideal gas against a piston whose pressure differs infinitesimally from the gas pressure, and phase transitions occurring at exactly the saturation temperature and pressure.

Characteristics of Irreversible Processes

An irreversible process generates entropy. Once it occurs, you cannot restore both the system and surroundings to their original states without leaving some change elsewhere.

• $\Delta S_{\text{universe}} > 0$ for every irreversible process.
• Sources of irreversibility include friction, unrestrained expansion, heat transfer across a finite temperature difference, mixing of different substances, and chemical reactions.
• Virtually all real-world processes are irreversible to some degree.
• Common examples: spontaneous heat transfer from a hot object to a cold one, free expansion of a gas into a vacuum ($Q = 0$ but $\Delta S > 0$), and diffusion of two gases into each other.