Revise Entropy and spontaneity for IB Chemistry (R1.4)

Entropy, symbol $S$, is a measure of how spread out the energy and matter of a system are. The more ways the particles and their quanta of energy can be arranged — the number of microstates, $W$ — the higher the entropy. A system left to itself moves towards the macrostate that can be realised in the greatest number of ways, which is why energy and matter tend to disperse rather than concentrate. 'Disorder' is a useful shorthand, but the precise idea is dispersal: a gas fills its container, a hot object shares its energy with a cold one, and a drop of dye spreads through water, all because the dispersed arrangement is overwhelmingly more probable. Standard molar entropies, $S^\ominus$, in J K⁻¹ mol⁻¹, are tabulated in the data booklet and are positive for every substance.

Many exam marks come simply from predicting whether $\Delta S$ is positive or negative and justifying it in one line. The reliable rule is to count the moles of gas on each side: gas contributes far more entropy than liquids or solids, so a change that makes more gas has $\Delta S > 0$, and a change that consumes gas has $\Delta S < 0$. When the moles of gas are unchanged, look to dissolving, state changes, or an increase in the total number of particles.

Because standard molar entropies are tabulated for individual substances, the entropy change of a reaction is a 'products minus reactants' sum, exactly like a Hess's-law enthalpy calculation. Each substance's $S^\ominus$ is multiplied by its coefficient in the balanced equation.

$\Delta S^\ominus = \sum S^\ominus(\text{products}) - \sum S^\ominus(\text{reactants})$

Enthalpy and entropy pull in their own directions, and Gibbs free energy combines them into one deciding quantity. At constant temperature and pressure, the change in Gibbs free energy is $\Delta G = \Delta H - T\Delta S$. Its sign is the verdict on spontaneity, because $\Delta G = -T\Delta S_{univ}$ — a negative $\Delta G$ is exactly a positive total entropy change of the universe, which the second law requires for a spontaneous change.

$\Delta G = \Delta H - T\Delta S$

Because the entropy term carries a factor of $T$, temperature can change the sign of $\Delta G$ — but only in two of the four sign combinations. When $\Delta H$ and $\Delta S$ agree (both favour going, or both oppose), temperature cannot overturn the verdict. When they disagree, temperature is the tie-breaker: raising it increases the weight of the $T\Delta S$ term until it dominates.

In the two temperature-dependent cases there is a crossover temperature at which the sign flips. It is found by setting $\Delta G = 0$, the point of equilibrium, and rearranging.

$\Delta G = 0 \ \Rightarrow\ 0 = \Delta H - T\Delta S \ \Rightarrow\ T = \frac{\Delta H}{\Delta S}$

Both quantities must be in matching energy units before dividing (both in J, or both in kJ). For an endothermic, entropy-increasing reaction ($\Delta H>0$, $\Delta S>0$) this $T$ is the temperature ABOVE which the reaction becomes spontaneous; for the enthalpy-driven case ($\Delta H<0$, $\Delta S<0$) it is the temperature BELOW which it stays spontaneous.

This is the showcase for a calculation topic. In Paper 2 the marks are analytic: each is tied to a specific line of working. A method mark (M) is earned for a correct process — the right substitution, the right rearrangement — even if a number is wrong; an accuracy mark (A) is earned for the correct value and DEPENDS on the method being present. Error-carried-forward (ECF) means a wrong number early on is only penalised once: later marks are awarded on your figures provided the method is sound. None of that protection exists unless the method is written down. Study how each of the five marks below attaches to one line.

Gibbs free energy is the hinge between the thermodynamics you have here and equilibrium later in the course: $\Delta G^\ominus$ is linked to the equilibrium constant, so a more negative $\Delta G^\ominus$ means an equilibrium lying further towards products. The same $\Delta G = \Delta H - T\Delta S$ explains why raising temperature can drive an endothermic reaction and shift its position of equilibrium, and it underpins electrochemistry, where cell potentials are Gibbs energies in disguise. Master the habit — predict the sign of $\Delta S$, match your units, substitute, judge the sign of $\Delta G$ — and the rest of physical chemistry becomes variations on a method you already own.