Entropy and Free Energy

A question asked by chemists is...

Why do some reactions take place of their own accord while others do not?

First, let's take a look at Entropy.

There is a natural tendency for physical and chemical systems to become more disordered¼

Think of your room at home. It has just been tidied for you and is very orderly. You may get up in a morning without making your bed. You read books and magazines leaving them open at various pages on your bed and across the floor. You change your clothes taking garments from your wardrobe but leaving those you have worn strewn about the room. Day by day the disorder gets worse. The repetition that existed amongst your books and magazine on shelves and your clothes hanging in a wardrobe is being lost. This tendency towards disorder is a natural one. There are many more ways books, magazines and clothes can be placed randomly than arranged in an orderly pattern. So, disorder is favoured. And, work must be done to restore order !

But what do we mean by order and disorder?

Order is characterised by repetition; disorder by a lack of patterns, an absence of organisation, by randomness and chaos.

Consider five gaseous atoms (He, Ne, Ar, Kr, Xe) in a vessel with two linked compartments (represented below). How many different ways can these atoms be distributed between the two compartments? Click here to show a possible arrangement (one of lowest entropy). Now click on the Entropy button to display other random arrangements.

Compartment 1
Compartment 2

There are 32 possible arrangements above. This is given by 2n, where n is the number of particles. With six gaseous atoms there are 64 arrangements between the two compartments:

Compartment 1Compartment 2Number of ways
601
516
4215
3320
2415
156
061

50 (15 + 20 + 15) of the 64 arrangements (78%) correspond to considerable spreading out. With just 100 atoms and two compartments, there are 1267650600228229401496703205376 ways they can distribute themselves, only two of which represent all the atoms together in one vessel. It now becomes even more convincing that we get even distribution of atoms and molecules by chance alone. So, the diffusion of gases can be explained in terms of probability.

The degree of disorder or randomness of a substance is measured by its entropy¼

The greater the disorder, the higher the entropy. The symbol for entropy, for no obvious reason, is S.

Entropy is a property of a system. As with enthalpy, we consider changes in entropy:

DS = Sfinal - Sinitial

Or

DS = SSproducts - SSreactants

DS depends only on initial and final states - it is not dependent on the path taken by the change to the system. For an increase in entropy DS has a positive value; for a decrease in entropy DS has a negative value.

The study of entropy is taken further...

Entropy and entropy changes refer to the number of ways of arranging atoms, molecules or ions in a system. The number of ways a given number of quanta of energy can be arranged between molecules makes another contribution. For example, four quanta of energy can be distributed between two molecules in 5 ways:

Molecule 1Molecule 2
40
31
22
13
04

Now raise the temperature of the system to provide more quanta. There are now more ways in which these can be shared out, and the more random or disordered the system becomes. This aspect too, along with the number of ways of arranging the molecules themselves, has to be included when calculating the entropies of substances, and entropy changes (DS) in chemical reactions.

Click here to read a short insight into the types of energy possessed by molecules.


Hide

An entropy value (unlike an enthalpy value) can be calculated for an element or compound...

Like enthalpy, H, the entropy of a substance depends upon conditions, such as temperature, pressure and the amount. Standard conditions are 298K, 100000 Pa (Nm-2), and 1 mole. Some Standard Molar Entropy values, S° are given below:

Substance S° (J K-1 mol-1)
Carbon (graphite) (s)5.7
Carbon (diamond) (s)2.4
Carbon dioxide (g)213.6
Ammonia (g)192.0
Hydrogen chloride (g)187.0
Ammonium chloride (s)95.0
Sodium chloride (s)72.1
Helium (g)126.0
Argon (g)154.7
Ethanol (l)160.7
Water (l)69.9
Water (g)188.7

We now need to address the question...

Why do some reactions take place of their own accord while others do not?

Let's be clear about something:
A spontaneous reaction is one that can 'possibly' take place, as opposed to a nonspontaneous process that cannot occur. However, a spontaneous process may not actually occur, or may be so slow that we think it does not occur. A spontaneous process is one that is thermodynamically feasible (energetically feasible) but may or may not be kinetically feasible. Thermodynamics tells us what cannot happen, but never guarantees that anything will happen.

Can we use change in Enthalpy, DH, to predict whether reactions will go of their own accord (occur spontaneously)? The answer is 'No', as we are familiar with both exothermic and endothermic reactions that take place readily in the laboratory.

Our reasoning with regard to Entropy gives us the idea that chemical reactions occur spontaneously if DS is calculated to be a positive value. But...

Try allowing some hydrogen chloride and ammonia gases to mix. Immediately, a finely powdered white solid (ammonium chloride, looking like white fumes), is formed.

The reaction has occurred spontaneously, though clearly there is a decrease in entropy when a solid product is formed from two gaseous reactants.

NH3(g) + HCl(g) ® NH4Cl(s)       DSsys = -284 J K-1 mol-1

Click here to see a calculation of DS for the above reaction from standard molar entropy values.

Why has this reaction taken place when DS is negative?

Here we have considered the entropy change involving only the reactants and product, that is, only of the system itself. In this case, DS should be written as DSsys. The reaction is exothermic (DH° = -176 kJ mol-1), and the energy given out increases the entropy of the surroundings, DSsurr. (Standard conditions will not be shown throughout.)

We need to consider the total entropy change, DStotal,

DStotal = DSsys + DSsurr

to decide if a reaction is spontaneous.

It can be shown (look up Trouton's Rule) that

DSsurr = -DH / T

(This calculates a positive value for DSsurr as the reaction is exothermic.)

Therefore,

DSsurr = - - 176000/298 = + 590 J K-1 mol-1

So,

DStotal = -284 + +590 = +306J K-1 mol-1

Thus, the reaction between HCl and NH3 is spontaneous because the total entropy change has a positive value.

The Second Law of Thermodynamics...

This is one way of expressing the Second Law of Thermodynamics. For a spontaneous change the total entropy must increase, DStotal > 0.

When DStotal = 0, there is no net change in either direction - the system has established equilibrium.

In every possible process the entropy of the universe increases...

In predicting the possible direction of a process, the entropy change of the universe is considered.

Specifically, we consider the entropy change in an 'isolated system'. This is a system that has no interaction with its surroundings. Nothing can enter or leave it. The entropy of an isolated system can never decrease. This is because energy cannot be transferred from the system to the surroundings.

But there is only one genuinely isolated system - it is the entire universe. It has no surroundings with which to interact. Given that there is no such thing as a truly isolated system, whenever the entropy of one system within the universe decreases, there must be another system whose entropy increases even more.

Gibb's Free Energy change, DG

Like DStotal, Gibb's Free Energy change, DG, allows us to decide if a reaction is spontaneous. DG has the advantage that it can be calculated entirely from measurements on the reacting system without regard to the surroundings. The equation is:

DG° = DH° - T DS°sys

Click here for an explanation.

The equation also shows that DG is dependent on temperature, so a chemical reaction that is not spontaneous at one temperature might be spontaneous at another.

Now refer back to the ammonia / hydrogen chloride reaction.

NH3(g) + HCl(g) ® NH4Cl(s)        DH° = - 176 kJ mol-1        DS°sys = - 284 J K-1 mol-1

DG° = - 176000 - (298 x - 284) = - 91.368 kJ mol-1

A negative value for DG indicates that the reaction is spontaneous. DG values can be used in Hess's Law calculations just like DH values.

What does the term 'free energy' relate to?

The chemical reaction above is exothermic, but not all of the energy evolved and absorbed by the surroundings, which increases the entropy of the surroundings (DSsurr), is needed to exceed the decrease in the entropy of the system (DSsys = -284 J K-1 mol-1) in this example. The energy left over is free energy available in any form for any use. It might simply warm up the surroundings (the chemicals themselves become warmer) or be obtained as electrical energy in an electrochemical cell. Click here for more explanation of 'free energy'.

Will a chemical reaction definitely take place if DG is negative, or if DStotal is positive?

First of all, a reaction will definitely not take place if DG is positive. If DG is negative, the reaction may still be immeasurably slow, effectively not taking place. The reaction may be thermodynamically unstable, but here it is kinetically stable. For a chemical reaction, chemical stability, or 'resistance to change', has two meanings; it must be considered in terms of both kinetic stability and thermodynamic stability. For a kinetically stable reaction, a large activation enthalpy may prevent the reaction from taking place at a reasonably measurable rate. DG is a thermodynamic quantity; it is concerned only with initial and final states, and tells us nothing about the height of the activation enthalpy 'barrier'. However, if DG is negative but the reaction is too slow, a catalyst might increase its rate. A catalyst works by providing an alternative route, or mechanism, for the reaction but of lower activation enthalpy. The diagram below shows catalysed and uncatalysed reaction profiles of a thermodynamically feasible reaction for which DG is negative.

Are there other quantities like DG ?

Gibb's Free Energy change, DG°, the equilibrium constant, K (Kc and Kp), and change in electrode potential, Dcell, for cell reactions, are all thermodynamic quantities that provide information about the thermodynamic feasibility of chemical reactions. For example, the more negative the value of DG°, and the larger the value of K, more of the reactants have formed products at equilibrium. A positive value calculated for Dcell predicts that the reaction concerned is thermodynamically feasible.

These equations show how DG° is related to K, and to Dcell (shown as E°cell):

DG° = - RT lnK

where R is the gas constant and T the absolute temperature.

DG° = - nFE°cell

where n is the number of electrons transferred and F the Faraday constant.