# 18.3 Gibbs Free Energy and the Relationship between Delta G, Delta H, and Delta S ## Gibbs Free Energy

Gibbs Free Energy is the thermodynamic quantity of a system that is the energy available to do work.  It is used to determine whether or not a reaction is spontaneous.  Simply put, spontaneous processes are those that occur 'naturally,' and nonspontaneous processes are those that do not.  What I mean by 'naturally' is that a reaction will occur in a system without the net influx of free energy from the surroundings.  For example, ice at 10oC and 1atm will melt spontaneously whereas ice at -10oC and 1atm will not.

What we observe is that during a spontaneous process a system will 'use up' some of its free energy and therefore the change in Gibbs free energy is negative (ΔG<0) for a spontaneous process.  Likewise the change in Gibbs free energy is positive (ΔG>0) for a nonspontaneous process and requires the input of free energy from the surroundings.  Finally, the change in Gibbs free energy is zero (ΔG=0) for a reaction that has reached equilibrium.  These are summarized in the table below.

MEANING OF ΔG VALUES
ΔG<0 Spontaneous
ΔG>0 Nonspontaneous
ΔG=0 At Equilibrium

## The Relationship between ΔG, ΔH, and ΔS

The change in Gibbs free energy (ΔG) for a system depends upon the change in enthalpy (ΔH) and the change in entropy (ΔS) according to the following equation:

ΔG = ΔH - TΔS

ΔGo = ΔHo - TΔSo

The relationship holds true under standard conditions or under non-standard conditions.  We can take away a few generalizations regarding when a reaction will be spontaneous (i.e. when ΔG<0).

A negative value for ΔH and a positive value for ΔS both contribute toward achieving a negative value for ΔG and a spontaneous reaction.  And for a reaction to even have a chance of being spontaneous at least one of these (negative ΔH or positive ΔS) must be true.

The first term in the calculation of ΔG is ΔH, the enthalpy change, and for many reactions/conditions this is the dominant term in the equation.  This is why we often anticipate that most exothermic reactions (negative ΔH) will be spontaneous and most endothermic reactions (positive ΔH) will not, but we cannot say this with absolute certainty.

The second term in the calculation of ΔG is -TΔS.  ΔS is typically significantly smaller than ΔH explaining why ΔH is often the dominant term in the equation.  But temperature is also a part of this term and this term, and ΔS specifically, have an increasing importance as the temperature is increased.

We can summarize the following regarding when a reaction is spontaneous.

• When you have both a negative ΔH and a positive ΔS, ΔG will always be negative and the reaction is spontaneous at all temperatures.
• When the opposite is true (positive ΔH and negative ΔS), ΔG will never be negative and a reaction will not be spontaneous at any temperature, or you could say that the reverse reaction is spontaneous at all temperatures.
• When ΔH and ΔS are both negative, ΔG will only be negative below a certain threshold temperature and we say that the reaction is only spontaneous at 'low temperatures.'
• When ΔH and ΔS are both positive, ΔG will only be negative above a certain threshold temperature and we say that the reaction is only spontaneous at 'high temperatures.'

These are summarized in the following table:

- + - Spontaneous at All Temperatures
+ - + Nonspontaneous at All Temperatures
- - + Spontaneous at Low Temperatures
+ + - Spontaneous at High Temperatures

## The Relationship between ΔH and ΔS

There is a relationship between ΔH and ΔS for a system at one of its phase change temperatures, (i.e. melting/freezing or boiling point) students are often required to know.  Take for example boiling water at 100oC.  At the boiling temperature you actually have liquid and gaseous water in equilibrium with each other.  As for any system at equilibrium ΔG=0 leading to the following derivation:

ΔG = ΔH - TΔS

0 = ΔH - TΔS

TΔS = ΔH It is from this last expression that undergraduate students are presented with equations that relate the freezing temperature to the ΔH and ΔS of fusion and the boiling temperature to the ΔH and ΔS of vaporization:  One could also rearrange the equation to solve for temperature which could be used to solve for a freezing or boiling point.  And for reactions in which ΔH and ΔS are either both negative or both positive this expression could also be used to solve for the threshold temperature below which or above which a reaction would be spontaneous.

One thing to keep in mind for calculations involving any of these equations is that ΔG and ΔH values are often reported in kJ/mol whereas ΔS values are typically reported in J/K.mol.  Make sure to convert so that all units are the same (both kJ or both J...either way) before performing any calculations.  Finally, all temperatures should be in Kelvin (the absolute scale) when performing calculations.

## Deriving the Equation for Gibbs Free Energy

ΔG = ΔH - TΔS

We've taken a long look at Gibbs Free Energy, its relationship to the change in enthalpy and the change in entropy of a process, and how it can be used to predict the spontaneity of a reaction, but how did Gibbs come up with this?  While not something the typical undergraduate is required to know I include it here for the curious mind.  Gibbs actually derived his equation for his newly coined "Gibbs Free Energy" specifically as a way to determine if/when a reaction is spontaneous.  He actually derived it from the 2nd Law of Thermodynamics which states the following:

For a spontaneous process the entropy change of the universe is positive.

We could also express the 2nd Law as follows:

For a spontaneous process, ΔSuniverse > 0

But there are two parts to the universe, the system and the surroundings, and we could express the 2nd Law one final time as follows:

For a spontaneous process, ΔSsystem + ΔSsurroudings > 0

This is where Gibbs started.  But measuring quantities for the surroundings is problematic as it includes all the rest of the universe outside of the system being investigated.  So Gibbs set out to devise a way to determine the spontaneity of a process based only upon thermodynamic properties of the system alone.  For this he needed to define ΔSsurroundings in terms of the system and substitute it back into the 2nd Law.  The change in entropy is defined as ΔS = qrev/T.  From this we can derive an expression for ΔSsurroundings:

ΔSsurroundings = ΔHsurroundings / T

But the enthalpy increase or decrease of the surroundings is due to the flow of enthalpy to or from the system, and therefore ΔHsurroundings and ΔHsystem are equal in magnitude but opposite in sign:  ΔHsurroundings = -ΔHsystem.  We can substitute this into our definition of ΔSsurroundings.

ΔSsurroundings = ΔHsurroundings / T = -ΔHsystem / T

This can now be substituted back into the 2nd Law of Thermodynamics.

ΔSsystem + ΔSsurroudings > 0

ΔSsystem - ΔHsystem / T > 0

Finally multiplying all terms by -T yields Gibbs Free Energy equation (remember that multiplying or dividing an inequality by a negative number changes the sign).

-TΔSsystem + ΔHsystem  < 0     rearranged     ΔHsystem - TΔSsystem < 0

Gibbs now had a condition for spontaneity that relied only on thermodynamic properties of the system and then coined it 'Gibbs Free Energy.'

ΔGsystem= ΔHsystem - TΔSsystem

And therefore we have derived from the 2nd Law of Thermodynamics:

For a spontaneous process, ΔHsystem - TΔSsystem < 0

For a spontaneous process, ΔGsystem < 0

And there you have it; Gibbs had devised a method of predicting if/when a process is spontaneous based upon thermodynamic properties of the system alone.