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Part 3: ΔS Entropy Changes and ΔG Free Energy Changes

This page introduces the student to the concept of entropy and is fundamental as to why a physical or chemical change is possible. The entropy of a system does seem at first sight to be about how chaotic or random it seems to be BUT it is more to do with how many ways it is possible for a system to exist i.e. the what is the state of the system in which enables its energy to be distributed in the maximum number of ways. Its not an east concept to come to terms with and the page starts by pointing out the limitations of enthalpy changes in predicting whether a reaction will occur or not?

Energetics-Thermochemistry-Thermodynamics Notes INDEX

3.1 An Introduction to Entropy

 Entropy and the direction of physical change or chemical change

Why do chemical reactions occur? Why does magnesium react with hydrochloric acid? Why doesn't hydrogen react with magnesium chloride solution? Why does ice melt at 273 K (0oC)?

These are questions not concerned with the rates of change (kinetics), even though this may have a bearing on their apparent spontaneity, but the above questions have everything to do with the energy changes which are possible i.e. what changes are feasible and why? and what is the 'energetic' driving force of these changes?

3.1a Events that happen, are the ones most likely to happen i.e. the most probable outcome!

This sounds easy? sounds logical? obvious? BUT is it always obvious? Hmm! I'm afraid not.

For most exothermic reactions the very negative enthalpy change is usually indicative that the reaction will occur if initiated in some way

i.e. just mixing chemicals e.g. mixing an acid and alkali,

H+(aq) + OH(aq) ==> H2O(l)  ΔHθneut,298K = –57.1 kJ mol–1

or applying a means of ignition to a combustible mixture e.g. burning methane,

C3H8(g) + 5O2(g) ==> 3CO2(g) + 4H2O(l)  ΔHθcomb,298K(propane) = –2219 kJ mol–1

and frankly, you would be a bit surprised if these reactions didn't happen.

At one time it was thought that only exothermic reactions where spontaneous and the more heat released to the surroundings the more likely the change would take place BUT although in the minority, there are plenty of spontaneous endothermic changes in which heat is absorbed from the surroundings and a temperature fall can be recorded with a thermometer e.g.

(i) The dissolving of the salt ammonium nitrate in water

NH4NO3(s) + aq ==> NH4+(aq) + NO3(aq)  ΔH = +25.8 kJ mol–1

(ii) The reaction between hydrated cobalt(II) chloride and sulfur dichloride oxide (thionyl chloride) which should be done in a fume cupboard due to its spectacular gas producing performance BUT not due to its exothermicity!

CoCl2.6H2O(s) + 6SOCl2(l) ==> CoCl2(s) + 6SO2(g) + 12HCl(g)  ΔH = +? kJ mol–1

ΔHθreaction,298 = ∑ΔHθf,298(products) – ∑ΔHθf,298(reactants)

ΔHθreaction,298 = {ΔHθf(CoCl2(s)) + 6xΔHθf(SO2(g)) + 12xΔHθf(HCl(g))}

{ΔHθf(CoCl2.6H2O(s)) + 6xΔHθf(SOCl2(l))}

ΔHθreaction = {–326 –6x297 –12x92.3} – {–2130 –6x206} = +150 kJ mol–1

This rapid reaction is very spontaneous and very cooling i.e. it is clearly and endothermic reaction! The huge volumes of extra gas molecules lead to a very large increase in entropy (a very +ΔS) which will explained in detail below – but remember this example.

(iii) However, contrast these two spontaneous reactions initiated at room temperature with the decomposition of limestone, whose is ΔH is +179 kJ mol–1, but only becomes feasible–spontaneous at over 900oC!

The thermal decomposition of calcium carbonate.

CaCO3(s) ==> CaO(s) + CO2(g)  ΔH = +179 kJmol–1

Unlike examples (i) and (ii), (iii) is not feasible at room temperature BUT will go spontaneously and commercially feasible at temperatures above 1000oC where the reaction is just as endothermic as at room temperature!

Clearly the feasibility of process or change cannot solely depend on the enthalpy change and in order to explain further you need to consider the concept of ENTROPY and some of its ramifications.

Concept: The entropy of a system is a measure of the number of ways a system can be arranged. This can been spatial arrangement–distribution of the particles or the distribution of all the quanta of energy internal to an atom, ion, molecule etc.

In Parts 1 Thermochemistry and Part 2. on the Born–Haber cycle we relied exclusively on the 1st Law of thermodynamics i.e. energy is conserved. In this section Part 3. we need get into the 2nd second law of thermodynamics states that the entropy of a system not at equilibrium will increase to give the maximum entropy state i.e. it will change to the most probable state. Its a bit abstract and takes a bit of getting used to and beware of simple explanations relying too heavily on relative order and disorder – though this idea has its place in coming to terms with the concept of entropy.

SO BEWARE – increase in entropy is often described as increasing chaos–disorder – but always try express your answers in terms of ... 'the more ways of arranging the outcome' – but this will only become clearer as you read further on. However in many ways it is conceptually useful at first to think of the greater the number of ways of ... the greater the entropy of the system and expressed in words as more disordered, spread out, mixed up or the more chaotic etc. etc.

3.1b The more ways an event can happen, the more probable is that event and the higher the entropy of a system

If there are more ways to arrange a 'system' the more likely is that 'system' situation will be attained. You may be talking about the number of possible ways the particles in a system can be arranged or the number of ways the quanta of energy can be distributed i.e. the internal energy of the particles in a system (atoms, ions, molecules) which may be in the form of occupied rotational, vibrational or electronic quantum levels – remember the possible levels are governed by quantum rules. Each possible way of arranging a system is often referred to as a microstate. A system will adopt the 'arrangement' that offers the maximum statistical probability of microstates.

You don't need to know this, but entropy is defined in a mathematical way as

S = klnW in which S = entropy, k = Boltzmann constant and lnW = the logarithm of the number of microstates possible for a system.

This equation is carved on the tombstone of Ludwig Boltzmann in recognition of his incredible intellectual contribution to the development of quantum physics and statistical thermodynamics. He committed suicide on Sept 5th 1906 after suffering bouts of depression and vilification from other minority physicists such as Ernst Mach who refused to accept the theories proposed by Boltzmann and others.

3.1c The entropy of a system not at equilibrium will always try to increase, determining the direction of change

(c) doc bThis is another way of expressing the 2nd law of thermodynamics. The use of the word equilibrium should be taken in its widest sense i.e. a system in equilibrium shows no net change on the macroscale.

e.g. gases will always diffuse into each other (e.g. bromine–air), the random motion of the molecules ensures that the most likely situation occurs i.e. multiple random distribution situations are possible and the outcome will be the most mixed up system possible. There is zero probability that they will unmix! In other words the outcome here is the situation that leads to the maximum possible ways of spatially arranging the particles.

Miscible liquid layers will mix in the same way as gases do, for exactly the same reason. However, unlike in gases, intermolecular forces can intervene to restrict the outcome. For example, on mixing the immiscible liquids oil and water, the two layers reform after shaking. Why? The strong hydrogen bonding between water molecules is stronger than any possible oil–water interaction.

immiscible liquid layers (c) doc b) VERSUS miscible liquids

(c) doc b) + (c) doc b) ==> (c) doc b)

3.1d The entropy of substances increases gas > liquid >> solid Why?

From left to right there is an increasing effect of inter–particle forces restricting the ways the particles or their kinetic energies can be distributed. In a gas the molecules are free to move randomly so there are endless permutations of spatial arrangement. In a liquid there is random movement but it is restricted by the intermolecular forces and clumps of molecules exist albeit for a tiny fraction of time before bombardment breaks them up – BUT clumps of molecules amount to tiny transient pockets of lower entropy. In a solid the particles can hardly move around so the spatial position possibilities are very limited.

Note: The above arguments only discuss spatial positioning and freedom of movement, but the distribution of energy e.g. kinetic is another contribution to the entropy of a substance and this factor is discussed in the next paragraph 3.1e. However in terms of the order/disorder solids are clearly highly ordered (short range and long range), liquids have a little short range order and gases are totally randomised at any given instant in time with virtually no inter–particle order at all. So in terms of possible 'arrangements' i.e. entropy, the general expected entropy order is not surprisingly is  gases > liquids > solids. This is a general trend for a range of substances and a specific trend for every individual substance. The entropy of a substance increases with temperature (see next section 3.1e and associated graph) and the most dramatic increases occur when there is a change in state (s) ==> (l) and (l) ==> (g).

See section Thermodynamics section 3.2 for some examples of entropy values

(c) doc b (c) doc b (c) doc b

The production of a gas in reactions is quite a driving force for 'unfavourable' reactions e.g. thermal decompositions like limestone ==> lime OR the cracking of alkanes are both very endothermic because the net increase in gas molecules considerably increases the entropy of a system driving the reaction in the decomposition direction so at higher temperatures many endothermic reactions become feasible. More on feasibility later.

3.1e Heat Capacity, energy distribution and entropy

The term specific heat capacity indicates quantitatively the energy required to increase a specified unit of mass (g or kg) of a substance by a specific unit of temperature (K or oC). The specific heat capacity at constant pressure is denoted by Cp. For water it is 4.2 J g–1 K–1 (or 4200 J kg–1 K–1), and the former is more convenient to use in laboratory calorimeter experiment calculations.

What happens when you heat a substance? Why does the entropy of a substance increase with increase in temperature? See diagram on the left. Tm = melting point and Tb = boiling point. At each change of state there is a dramatic rise in the entropy S of a substance and in between a steady increase in the entropy of a substance.

The temperature of a substance depends on the kinetic energies of the particles – that is the motion of the particles or its component atoms. The more energetic the particles in terms of KE the higher the temperature and the 'hotter' it feels! So, how does this relate to the concept of entropy?

Notice that the entropy graph has its origin at 0,0 (S = 0 when T = 0). Theoretically at 0K all vibrations or rotations etc. have stopped and if a perfect crystal was formed its entropy is zero for this sole possible 'frozen' microstate.

The 3rd Law of Thermodynamics states that at absolute zero all perfect crystals have zero entropy.

S = klnW = 0 if W = 1 possible microstate

Molecules possess three forms of kinetic energy:

Translation – moving from one place to another as in a gas or liquid (virtually no translation in solids).

Rotation – the molecule or a grouping in the molecule spinning around.

Vibration – all atoms constituting bonds vibrate in some way either by stretching–compression along the bond axis or bending–relaxing movement.

All of these forms of energy are quantised, that is, only specific KE energies of translation, rotation and vibration are allowed and the more energy absorbed in raising the temperature of a substance the more of the higher quantum levels of KE are accessible i.e. more ways in which the energy can be distributed i.e. an entropy increase. Note that we are now talking about the distribution of energy and not just the possible spatial arrangements and freedom to move around.

The order of these three types of quantum level is vibrational > rotation > translational

The translational quantum levels are so close together that virtually any measurable KE or velocity is likely to be observed – but it does seem strange to think of this motion in a quantised way. Although not required at advanced level, it is possible to show by direct experiment that freely moving electrons, atoms or molecules behave as a wave as well as particle – but the quantum world is 'wacky one' so don't expect the reality of our macro physical world to match with the quantum world. However, the quantisation of vibrational and rotational levels shouldn't present a conceptual problem – just think that the vibrations and rotations can only occur at specific frequencies determined by the quantum number rules – which you don't have to worry about. Also, remember microwave absorption is due to rotational quantum level changes and infra–red absorption is due to vibrational level changes.

As you heat a substance more and more of these 'kinetic energy' quantum levels can be accessed as the temperature of the substance increases, therefore more ways to distribute the energy, therefore a higher entropy state is obtained.

3.1f Electronic energy levels and entropy

If you continue to heat a substance eventually the kinetic energy of collisions–vibrations etc. is sufficient to cause electronic energy quantum level changes. Molecules can be raised to an 'excited' state when an electron is raised to a higher quantum level, and, if raised sufficiently highly, bond breaking will occur. This does not necessarily increase the temperature of substance because electronic quantum level changes do not affect the motion of the molecules i.e. the kinetic energy of the molecules. However, what it does mean is that yet more 'energy levels' are available in which to distribute the energy absorbed i.e. yet again this will lead to a higher entropy level being attained.

So the complete order of the four types of quantum level is electronic > vibrational > rotation > translation and, depending on the physical state and temperature of a substance, there is continuous interchange of energy between the possible energy levels e.g. via particle collision – remember energy cannot be created or destroyed, but can be changed in 'form' or 'distribution'.

Incidentally, excited molecules can be produced by e.g. uv light, but if no bond fission occurs, the molecules 'relax' by changing the absorbed electronic quantum level energy into some form of kinetic energy – vibrational, rotational and translational, so in the end the temperature of the substance is increased. Another example is holding your hand in sunlight – it becomes hotter because the molecules of your skin absorb IR and higher vibrational levels are accessed – in the end the molecules relax and the absorbed energy ends up as translational kinetic energy – but either way, the temperature of your skin increases as does its entropy!

There is a further conceptual quantum level complication because these types of quantum levels, although specific for a given bond or molecule etc. are not independent of each other. Each electronic level has its own set of associated vibrational levels, each vibrational level has its own set of associated rotational levels and each rotational level has its own set of translational levels. BUT don't worry about this! the point is that when a system–substance absorbs energy, the energy will be distributed in the maximum possible number of ways i.e. to attain the highest possible entropy state.

3.1g A summing up so far before we get into examples and then calculations!

We have considered the concept of the entropy state of a substance in terms of :–

A substance/system not at 'equilibrium' (i.e. a 'no net change state') will try to attain the highest entropy state by physical or chemical change. We can envisage the entropy of a substance in terms of the spatial distribution of the particles in the substance and the distribution of the energy of individual particles between the available translational, rotational, vibrational and electronic energy levels of each particle.

Therefore, entropy is a measure of the number of ways particles can be spatially distributed AND the number of ways the quanta of energy can be distributed i.e. the way the energy is 'arranged' in the system.

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