What entropy is really measuring

Entropy, S, is a state property. It measures how widely matter and energy are dispersed in a system; its SI unit is joule per kelvin, J K⁻¹. In chemistry, we usually work with standard molar entropy, S⦵, the entropy per mole of a substance in its standard state. Its unit is J K⁻¹ mol⁻¹.

“More disordered means higher entropy” is a handy classroom shortcut, but don’t treat it as the definition. A better way to think about entropy is the number of possible arrangements. When particles and energy can be spread out in more ways, entropy is higher. So, under the same conditions, gases have higher entropy than liquids, and liquids have higher entropy than solids. Gas particles can occupy many more positions and move much more freely.

A system is the chemical or physical process being studied. The surroundings are everything outside that system that can exchange energy or matter with it. The total entropy change is written as ΔSₜₒₜₐₗ = ΔSₛᵧₛₜₑₘ + ΔSₛᵤᵣᵣₒᵤₙdᵢₙgₛ, where ΔSₜₒₜₐₗ is the total entropy change for the process, in J K⁻¹, or J K⁻¹ mol⁻¹ when stated per mole of reaction. ΔSₛᵧₛₜₑₘ is the entropy change of the reacting system, and ΔSₛᵤᵣᵣₒᵤₙdᵢₙgₛ is the entropy change of the surroundings. A spontaneous change increases total entropy; at equilibrium the total entropy is no longer increasing.

Predicting the sign of ΔS

For a physical change, focus on freedom of movement. Melting, vaporization and sublimation usually increase entropy. Freezing, condensation and deposition usually decrease entropy. Dissolving a solid often increases entropy because particles spread through the solvent, although cases with highly ordered solvent shells can be more subtle.

For a chemical change, look first at the states of matter, then at the balanced equation. Gases dominate entropy predictions because gaseous particles have far more possible positions than particles in solids or liquids. If the number of moles of gas increases, ΔS is usually positive. If the number of moles of gas decreases, ΔS is usually negative. If no gas is involved, compare how dispersed the particles become, for example through dissolving, dissociation or formation of more separate particles.

Balanced coefficients matter. One mole of solid forming two moles of gas gives a large entropy increase. Several moles of gas forming fewer moles of gas is usually an entropy decrease, even if liquid products are also formed.

Calculating standard entropy changes

Standard conditions are agreed reference conditions used for tabulated thermodynamic data, with substances in their standard states. Entropy is a state function, so the route taken does not matter; only the initial and final states matter.

For a reaction, ΔS⦵ = ΣS⦵(products) − ΣS⦵(reactants), where ΔS⦵ is the standard entropy change for the reaction as written, in J K⁻¹ mol⁻¹. Σ means “sum of”, and each S⦵ value is multiplied by its balanced equation coefficient before subtracting. Standard entropy values are provided in the data booklet.

When you calculate, keep the physical state attached to the formula. S⦵ for H₂O(g) is not the same as S⦵ for H₂O(l), because gas and liquid water have different energy dispersal and particle freedom. A sensible estimate of the sign before calculating is a good way to catch arithmetic slips.

The perfect crystal at 0 K

The Structure 1.1 link is the third law idea. A perfect crystal is a solid in which every particle occupies a regular, repeating lattice position with no defects. At 0 K, a perfect crystal is predicted to have zero entropy because there is only one possible arrangement of the particles and their energy in the lowest-energy state. In the language of this topic: there is no additional dispersal to count.

Why Gibbs energy is needed

Enthalpy by itself cannot tell us which way a chemical change will go. Many exothermic reactions are spontaneous, but not all of them. Some endothermic processes happen spontaneously because they produce a large increase in entropy. So we use one quantity that brings together enthalpy change, entropy change and temperature.

Gibbs energy, G, is a thermodynamic state function that shows the energy available to do useful work at constant temperature and pressure; its unit is joule, J. The change in Gibbs energy, ΔG, is the Gibbs energy change for a process; in this topic it is usually reported per mole of reaction in kJ mol⁻¹.

Under standard conditions, ΔG⦵ = ΔH⦵ − TΔS⦵, where ΔG⦵ is the standard Gibbs energy change (kJ mol⁻¹), ΔH⦵ is the standard enthalpy change (kJ mol⁻¹), T is the absolute temperature (K), and ΔS⦵ must be used in kJ K⁻¹ mol⁻¹ in this equation.

That last unit detail is the one I underline on the board. The data booklet gives ΔS⦵ values in J K⁻¹ mol⁻¹, while ΔH⦵ and ΔG⦵ are normally in kJ mol⁻¹. If the other quantities are in kJ mol⁻¹, divide entropy changes by 1000 before multiplying by temperature.

Rearranging the equation

You can use the same equation to find any unknown term. For example, ΔH⦵ = ΔG⦵ + TΔS⦵, and ΔS⦵ = (ΔH⦵ − ΔG⦵) / T. The algebra is straightforward; the chemistry is in keeping the units consistent and reading the sign correctly.

A graph of ΔG⦵ against T is useful because the equation is in the form of a straight line. The vertical intercept is ΔH⦵, and the gradient is −ΔS⦵. If the line crosses ΔG⦵ = 0, that temperature marks the switch between spontaneous and non-spontaneous behaviour under standard conditions.

What temperature does in the equation

Temperature changes the size of the entropy term, TΔS⦵. At low temperature, ΔH⦵ may dominate. At high temperature, the entropy term can dominate. That is why some processes are spontaneous only above a certain temperature, while others are spontaneous only below a certain temperature.

Interpreting ΔG

A spontaneous change is a process that proceeds in the stated direction under the given conditions without continual external driving; it may go to completion, or it may reach equilibrium. Spontaneous doesn’t mean fast. A reaction can be thermodynamically favourable but kinetically slow because it has a large activation energy.

At constant pressure, the sign of ΔG shows the thermodynamic direction:

• ΔG < 0: the forward process is spontaneous under those conditions.
• ΔG = 0: the system is at equilibrium; there is no net driving force.
• ΔG > 0: the forward process is non-spontaneous; the reverse process is spontaneous.

ΔG combines two entropy effects. One is direct: the entropy change of the chemicals themselves as bonds, states and particle numbers change. The other comes from the surroundings, through heat transfer. In an exothermic reaction, heat released to the surroundings tends to increase the surroundings’ entropy. In an endothermic reaction, heat absorbed from the surroundings tends to decrease it.

This is why an endothermic reaction can still be spontaneous. If the system entropy increases enough, that direct entropy gain can outweigh the entropy decrease of the surroundings. A reaction between two solids that produces gas and aqueous ions is a good example: matter becomes much more dispersed, even though heat is absorbed.

Combining signs of ΔH and ΔS

Learn the four sign combinations, because they let you predict temperature dependence before doing a calculation.

Four ΔH° and ΔS° sign combinations and their effect on ΔG° and spontaneity.

If ΔH⦵ is negative and ΔS⦵ is positive, ΔG⦵ is always negative: exothermic and more dispersed is the most favourable combination. If ΔH⦵ is positive and ΔS⦵ is negative, ΔG⦵ is always positive: endothermic and less dispersed is unfavourable at all temperatures.

The remaining two cases depend on temperature. If both ΔH⦵ and ΔS⦵ are positive, the reaction becomes spontaneous at high temperature because TΔS⦵ eventually becomes larger than ΔH⦵. If both ΔH⦵ and ΔS⦵ are negative, the reaction is spontaneous at low temperature but becomes non-spontaneous at high temperature, since subtracting a negative entropy term makes ΔG⦵ larger.

Finding the temperature where spontaneity changes

At the boundary between spontaneous and non-spontaneous behaviour, ΔG⦵ = 0. So 0 = ΔH⦵ − TΔS⦵, giving T = ΔH⦵ / ΔS⦵, where T is the threshold temperature (K), ΔH⦵ is in kJ mol⁻¹, and ΔS⦵ must be in kJ K⁻¹ mol⁻¹.

This threshold only has physical meaning when the sign pattern allows a crossing. For ΔH⦵ > 0 and ΔS⦵ > 0, the reaction is spontaneous above the calculated temperature. For ΔH⦵ < 0 and ΔS⦵ < 0, it is spontaneous below the calculated temperature.

Link to electrochemical data

Electrochemical data can predict spontaneity too. For a cell reaction under standard conditions, ΔG⦵ = −nFE⦵cell, where n is the number of moles of electrons transferred per mole of reaction (mol e⁻ mol⁻¹ reaction), F is the Faraday constant (C mol⁻¹), and E⦵cell is the standard cell potential (V). A positive E⦵cell gives a negative ΔG⦵, so the cell reaction is spontaneous as written. It’s the same thermodynamic test, using electrical data instead of enthalpy and entropy data.