Gibbs Free Energy and Spontaneity of Reactions Chemistry Tutorial

Key Concepts

• Gibbs free energy is also known as:

  ⚛ free energy

  ⚛ Gibbs free enthalpy

  ⚛ free enthalpy

• Gibbs free energy (free energy) is given the symbol G
• Gibbs free energy (free energy), G, of a chemical system is defined as:

  G = H -TS

  H = enthalpy
  T = temperature
  S = entropy

• ΔG represents the change in Gibbs free energy for a chemical system at constant temperature and pressure

  ΔG = ΔH -TΔS

  ΔH = enthalpy change of the system
  T = temperature of the system
  ΔS = entropy change of the system

• For a chemical system at constant temperature and pressure, the reaction is

  ⚛ spontaneous if ΔG < 0

  (the sign of ΔG is negative)

  ⚛ nonspontaneous (not spontaneous) if ΔG > 0

  (the sign of ΔG is positive)

  ⚛ at equilibrium if ΔG = 0

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Defining Gibbs Free Energy, G

Whether or not a chemical reaction or a physical change is spontaneous depends on the total entropy change of the entire system.
The total entropy change of the entire system includes both the entropy of the chemical system under investigation AND also the entropy of the surroundings.
That is:

While it is often possible to estimate the entropy change for a chemical system, ΔS_{chemical system}, it is almost impossible to determine the entropy change for the surroundings, ΔS_surroundings, given that the surroundings are actually the entire universe!
The problem of estimating the entropy change for the entire universe can be overcome by restricting ourselves to a consideration of chemical processes that occur only at constant temperature and constant pressure.

At constant temperature, the entropy change of the surroundings (ΔS_surroundings) depends on the:

• amount of heat energy absorbed by the surroundings when there is a change in the chemical system being investigated
• temperature (T) at which heat is transferred to the surroundings from the chemical system being investigated

We can write an equation to represent this entropy change in the surroundings at constant temperature as shown below:

At constant pressure, the amount of heat absorbed by the surroundings equals -q where q equals the amount of heat absorbed by the chemical system.

q_system = heat absorbed by chemical system

-q_system = heat absorbed by surroundings (that is, heat is transferred from surroundings to be absorbed by the chemical system)

Therefore, we can write:

Note that if the chemical system:

• gains heat (absorbs heat), q is a positive number, so -q is a negative number
• loses heat (releases heat), q is a negative number, so -q is a positive number

At constant pressure, q equals the enthalpy change for the system:

q_system = ΔH_system

At constant pressure, the amount of heat absorbed by the surroundings = -q_system
So, the amount of heat absorbed by the surroundings = -ΔH_system

At constant temperature AND pressure, we can write the following equation to represent the change in entropy for the surroundings:

Note that if the chemical process is:

• exothermic, ΔH_system is negative, -ΔH_system is positive, and ΔS_surroundings is positive, entropy of the surroundings increase
• endothermic, ΔH_system is positive, -ΔH_system is negative, and ΔS_surroundings is negative, entropy of the surroundings decrease

Now we can substitute our expression for ΔS_surroundings into our original equation for determining the change in total entropy:

Multiply throughout by -T

Which can be rearranged in give:

-TΔS_total = ΔH_{chemical system} - TΔS_{chemical system}

We define the Gibbs free energy of a chemical system, or free energy of a chemical system, G_system, as:

G_system = H_system - TS_system

where:

H_system = enthalpy of the chemical system

T_system = temperature of the chemical system

S_system = entropy of the chemical system

G_system, H_system, and S_system depend only on the state of a system.
If there is a change of state, say from state 1 to state 2, then:

G_{state 2} - G_{state 1} = H_{state 2} -H_{state 1} - (T_{state 2}S_{state 2} - T_{state 1}S_{state 1})

That is:

ΔG_system = ΔH_system - (T_{state 2}S_{state 2} - T_{state 1}S_{state 1})

At constant temperature, T_system = T_{state 2} = T_{state 1}
So we can write the equation for the change in Gibbs free energy of the system (G_system) as:

ΔG_system = ΔH_system - (T_systemS_{state 2} - T_systemS_{state 1})

ΔG_system = ΔH_system - T_system(S_{state 2} - S_{state 1})

ΔG_system = ΔH_system - T_systemΔS_system

Compare this with our equation for determining the total entropy change (ΔS_total) for a chemical process occurring at constant temperature and pressure:

-TΔS_total = ΔH_{chemical system} - -TΔS_{chemical system}

The change in Gibbs free energy for the system (ΔG_system) can be equated with the total entropy change multiplied by -temperature:

ΔG_system = -TΔS_total

Using Gibbs Free Energy (G) to Determine if a Chemical Reaction is Spontaneous or Nonspontaneous

According to the Second Law of Thermodynamics, the entropy of the universe increases, that is, the total entropy (ΔS_total) increases.

For a spontaneous change going from state 1 to state 2 the entropy of state 2 (S_{state 2}) must be greater than the entropy of state 1 (S_{state 1})

S_{state 2} > S_{state 1}

So the change in total entropy (ΔS_total) going from state 1 to state 2 spontaneously must be positive:

ΔS_total = S_{state 2} - S_{state 1}

ΔS_total > 0

For a chemical process occurring at constant temperature and pressure, this means that the change in the Gibbs free energy of the system (ΔG_system) must be negative for a spontaneous reaction:

ΔG_system = -TΔS_total

ΔS_total is positive for a spontaneous reaction

ΔG_system = negative

ΔG_system < 0

A process in which the total entropy decreases (ΔS_total < 0) would be nonspontaneous.

At constant temperature and pressure, the change in Gibbs free energy for this nonspontaneous process would be positive:

ΔG_system = -TΔS_total

ΔS_total is negative for a nonspontaneous reaction

ΔG_system = positive

ΔG_system > 0

For a system at equilibrium there is no change in total entropy (ΔS_total = 0)

At constant temperature and pressure, the change in Gibbs free energy for this process at equilibrium would be 0:

ΔG_system = -TΔS_total

ΔS_total = 0 for a system at equilibrium

ΔG_system = -T × 0

ΔG_system = 0

The table below summarises the values for the change in Gibbs free energy for spontaneous reactions, nonspontaneous reactions and systems at equilibrium at constant temperature and pressure:

Worked Example of Gibbs Free Energy and Spontaneity of Reactions

Question:

Iso-octane (2,2,4-trimethylpentane), C₈H₁₈, is a component of petrol (gasoline), a fuel commonly used in internal combustion engines.

The complete combustion of iso-octane in an internal combustion engine is described by the balanced chemical equation given below:

C₈H_18(g) + 12½O_2(g) → 8CO_2(g) + 9H₂O_(g)

At 25°C (298 K) and 1 atmosphere pressure (101.3 kPa), the combustion of 1 mole of iso-octane releases 5109 kJ heat energy.

Predict the sign of the change in Gibbs free energy (ΔG) for this reaction.

Solution:

(Based on the StoPGoPS approach to problem solving.)