Entropy and spontaneity

Diamond has a higher Gibbs free energy than graphite, so under standard conditions it "should" turn into graphite by itself. Yet a diamond ring passed down in a family lasts for generations without changing. Knowing that a reaction is spontaneous tells you nothing about how fast it is. Gibbs free energy and activation energy answer two different questions, and mixing them up is one of the most common mistakes in this topic.

Entropy: the dispersal of energy

Drop a bit of ink into a glass of water and it spreads out evenly. You never see the ink molecules gather themselves back into one corner. What quantity explains why natural processes have a preferred direction?

Gibbs free energy and spontaneity

A cold pack dissolves ammonium nitrate in water and the pack gets icy cold. The process is endothermic ($\Delta H > 0$), yet it runs by itself and goes to completion. If reactions do not simply "roll downhill" in energy, what quantity balances heat against disorder?

The table below assumes standard conditions and that $\Delta H^\circ$ and $\Delta S^\circ$ stay constant over the temperature range, with no change of state.

The four cases

$\Delta H^\circ$	$\Delta S^\circ$	$\Delta G^\circ$	Spontaneous?
$< 0$	$> 0$	always $< 0$	at all temperatures
$> 0$	$< 0$	always $> 0$	never
$< 0$	$< 0$	depends on $T$	at low $T$
$> 0$	$> 0$	depends on $T$	at high $T$

Why? As $T$ rises, the term $-T\Delta S^\circ$ becomes more important, so the sign of $\Delta S^\circ$ eventually decides the sign of $\Delta G^\circ$.

The minimum temperature for spontaneity

$\mathrm{CaCO_3(s)} \rightarrow \mathrm{CaO(s)} + \mathrm{CO_2(g)}$, with $\Delta H^\circ = +178.3\ \text{kJ/mol}$ and $\Delta S^\circ = +160.5\ \text{J/(mol·K)}$.

Both $\Delta H^\circ$ and $\Delta S^\circ$ are positive, so the reaction is spontaneous only at high enough temperature.

Set $\Delta G^\circ = 0$ to find the threshold. Convert units first: $\Delta H^\circ = 178,300\ \text{J/mol}$.

$T = \dfrac{\Delta H^\circ}{\Delta S^\circ} = \dfrac{178,300}{160.5} \approx 1,111\ \text{K}$. Above this temperature $\Delta G^\circ < 0$ and the limestone decomposes. This is why industrial lime kilns run at temperatures well above 1,100 K.

Calculate $\Delta G^\circ$ at $298\ \text{K}$ and say whether the reaction is spontaneous at room temperature.

The graph shows $\Delta G^\circ = 40 - 0.10\,T$ (in kJ/mol) for a hypothetical endothermic reaction. Where the line crosses zero, the reaction becomes spontaneous.

Energy efficiency of heating water

You heat water for pasta every week. Does all the energy you pay for end up in the water? Energy is never destroyed, but it spreads out, and entropy explains why some of it always escapes as low-grade heat. For how to calculate heat, see the page on energy, heat and work (TERMO-PRINCIPIO).

Practice★★★

Topic challenge. Is your kitchen efficient? Evaluate the thermodynamics of heating water in your home in the context of the energy transition. 1. Domestic calorimetry:…

Topic challenge. Is your kitchen efficient? Evaluate the thermodynamics of heating water in your home in the context of the energy transition.

Domestic calorimetry: Heat $0.5\text{ kg}$ of water in a saucepan for $3\text{ min}$. Measure $T_i$ and $T_f$ (or take the boiling temperature) and calculate the useful heat absorbed using $Q = m \times c_e \times \Delta T$.
Hess, bonds and the first law: Calculate the molar enthalpy of combustion ($\Delta H^\circ$) of the gas used (butane or methane) by applying Hess's law or bond energies. Estimate the expansion work ($W = -\Delta n_g R T$) of the gases produced and deduce $\Delta U$. Determine the thermal efficiency of your stove by estimating the energy supplied from the burner's rated power (or the gas consumed) during the $3\text{ min}$.
Entropy and spontaneity: Look up the ambient temperature of your city on the National Weather Service website. Using tabulated standard thermodynamic data, calculate $\Delta S^\circ$ and $\Delta G^\circ$ at that temperature and explain the spontaneity of the process.
Technical proposal: Write a reasoned report (one page) with comparative tables, justifying the heat losses and suggesting practical energy-saving measures.

Assessment criteria:

Rigor in setting up and calculating thermodynamic variables ($Q$, $W$, $\Delta H^\circ$, $\Delta S^\circ$ and $\Delta G^\circ$).
Handling and consistency of the home experimental data and the official meteorological data (National Weather Service).
Capacity for critical analysis when interpreting the thermal efficiency and the spontaneity of the system.

Solve

$\Delta G$, $\Delta G^\circ$ and the equilibrium constant

Put only reactants in a flask for a reaction with $\Delta G^\circ > 0$. Some product still forms straight away, even though the reaction looks "unfavourable" on paper. Why does the real driving force change with the mixture, and exactly when does it stop?

Back to the start: diamond is thermodynamically unstable relative to graphite ($\Delta G^\circ < 0$), but the activation energy is huge, so the change is too slow to see. Your family's diamond ring is safe.

Summary

Entropy, $S$, measures the dispersal of energy among possible arrangements. It rises with disorder, with more particles and, above all, when a substance becomes a gas.
At constant $T$ and $P$, $\Delta G = \Delta H - T\Delta S$: negative means spontaneous, positive means the reverse is favored, and zero means equilibrium.
The signs of $\Delta H^\circ$ and $\Delta S^\circ$ give four cases. When temperature matters, the threshold is $T = \Delta H^\circ / \Delta S^\circ$ (use consistent units).
Spontaneity (thermodynamics) and speed (kinetics) are independent: a negative $\Delta G^\circ$ does not mean you will see the reaction happen.
$\Delta G^\circ = -RT\ln K$ links standard Gibbs free energy to the equilibrium constant, and $\Delta G = \Delta G^\circ + RT\ln Q$ gives the direction a mixture will move.

Let's practice! 0 medium exercises in this section

Let's practice! 1 hard exercises in this section

★★★ Topic challenge. Is your kitchen efficient? Evaluate the thermodynamics of heating water…

See all the exercises in this section