6.3 Entropy and the second law of thermodynamics

Entropy and the Second Law of Thermodynamics

Entropy and thermodynamics

Entropy ($S$) is a thermodynamic state function that measures the degree of disorder or randomness in a system. A gas has high entropy because its molecules move freely in all directions, while a solid has low entropy because its particles are locked in a rigid structure. Liquids fall somewhere in between.

The Second Law of Thermodynamics states that the total entropy of the universe always increases for any spontaneous process.

• In an isolated system, spontaneous processes move in the direction of increasing entropy (think of ice melting at room temperature).
• Reversible processes produce no change in total entropy. These are idealized processes where the system stays in near-equilibrium at every step.
• Irreversible processes (which includes every real process) always produce a net increase in total entropy. Heat flowing from a hot object to a cold one is a classic example.

Entropy calculations for processes

There are two key equations for calculating entropy changes, and which one you use depends on the type of process.

For chemical reactions, use standard molar entropies from a reference table:

$\Delta S^\circ = \sum nS^\circ_\text{products} - \sum nS^\circ_\text{reactants}$

Here $n$ represents the stoichiometric coefficients. You multiply each substance's standard molar entropy ($S^\circ$, in units of J/(mol·K)) by its coefficient, then subtract the reactant total from the product total.

For example, consider the combustion of methane:

$CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(g)$

You'd calculate $\Delta S^\circ$ as:

$\Delta S^\circ = [S^\circ(CO_2) + 2 \cdot S^\circ(H_2O)] - [S^\circ(CH_4) + 2 \cdot S^\circ(O_2)]$

Plug in the tabulated values, and the arithmetic gives you the entropy change for the reaction under standard conditions.

For phase transitions at constant temperature and pressure:

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

Here $\Delta H$ is the enthalpy change of the transition (in J) and $T$ is the temperature in Kelvin at which the transition occurs. This equation applies at the exact phase-change temperature, where the process is reversible. For instance, water melting at 273 K or vaporizing at 373 K at 1 atm.

A quick example: the enthalpy of fusion of water is 6.01 kJ/mol (6010 J/mol). At the melting point:

$\Delta S_{fus} = \frac{6010 \text{ J/mol}}{273 \text{ K}} = 22.0 \text{ J/(mol·K)}$

The positive sign confirms that melting increases entropy, which makes sense since liquid water is more disordered than ice.

Predicting entropy changes

You can often predict the sign of $\Delta S$ without doing any math by asking: does the process increase or decrease molecular disorder?

Positive entropy changes ($\Delta S > 0$):

• Melting (solid → liquid) and vaporization (liquid → gas)
• Dissolving a solid in a liquid (e.g., NaCl dissolving in water spreads ions throughout the solution)
• Reactions that produce more moles of gas than they consume
• Gas expansion into a larger volume

Negative entropy changes ($\Delta S < 0$):

• Freezing and condensation
• Crystallization (e.g., ice crystals forming from liquid water)
• Reactions that produce fewer moles of gas

When counting gas moles, focus on the gas-phase species specifically. If a reaction goes from 2 moles of gas on the reactant side to 3 moles of gas on the product side, expect $\Delta S > 0$, regardless of what's happening with solids or liquids.

Temperature also matters. At higher temperatures, molecules already have more kinetic energy, so adding a given amount of heat produces a smaller relative change in entropy. This is reflected in the $\Delta S = \frac{\Delta H}{T}$ equation: a larger $T$ in the denominator means a smaller $\Delta S$ for the same $\Delta H$.

Entropy and process spontaneity

A spontaneous process occurs on its own without continuous external intervention. The Second Law gives you the criterion: a process is spontaneous when the total entropy of the universe increases.

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

• If $\Delta S_\text{universe} > 0$, the process is spontaneous.
• If $\Delta S_\text{universe} = 0$, the process is at equilibrium (reversible).
• If $\Delta S_\text{universe} < 0$, the process is non-spontaneous in that direction (but it is spontaneous in the reverse direction).

A common point of confusion: a process can be spontaneous even if the system's entropy decreases, as long as the surroundings' entropy increases by a larger amount. For example, when crystals form from a supersaturated solution, the system becomes more ordered ($\Delta S_\text{system} < 0$), but the heat released to the surroundings increases $\Delta S_\text{surroundings}$ enough to make $\Delta S_\text{universe}$ positive overall.

You can calculate the entropy change of the surroundings using:

$\Delta S_\text{surroundings} = \frac{-\Delta H_\text{system}}{T}$

The negative sign is there because heat leaving the system enters the surroundings. An exothermic reaction ($\Delta H_\text{system} < 0$) releases heat, which increases the entropy of the surroundings.

All real processes are irreversible to some degree. They always produce a net increase in the universe's entropy and a decrease in the overall quality of usable energy. Combustion of fuel, heat flowing from a hot object to a cold one, and gas escaping from a punctured tire are all irreversible processes that can never fully "undo" themselves without changing something else in the universe.