Please do not block ads on this website.
No ads = no money for us = no free stuff for you!
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:
total entropy change 
= 
entropy change change for the chemical system 
+ 
entropy change of the surroundings

ΔS_{total} 
= 
ΔS_{chemical system} 
+ 
ΔS_{surroundings}

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:
ΔS_{surroundings} = 
heat absorbed by surroundings T 
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:
ΔS_{surroundings} = 
heat absorbed by surroundings T 
ΔS_{surroundings} = 
q_{system} T 
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:
ΔS_{surroundings} = 
ΔH_{system} T 
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:
change in total entropy 
= 
change in entropy for the chemical system 
+ 
change in entropy of the surroundings

ΔS_{total} 
= 
ΔS_{chemical system} 
+ 
ΔS_{surroundings}

ΔS_{total} 
= 
ΔS_{chemical system} 
+ 
ΔH_{chemical system} T

ΔS_{total} 
= 
ΔS_{chemical system} 
 
ΔH_{chemical system} T

Multiply throughout by T
T × ΔS_{total} 
= 
T × ΔS_{chemical system} 
 
T × ΔH_{chemical system} T

TΔS_{total} 
= 
TΔS_{chemical system} 
 
ΔH_{chemical system}

TΔS_{total} 
= 
TΔS_{chemical system} 
+ 
ΔH_{chemical system}

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_{system}S_{state 2}  T_{system}S_{state 1})
ΔG_{system} = ΔH_{system}  T_{system}(S_{state 2}  S_{state 1})
ΔG_{system} = ΔH_{system}  T_{system}ΔS_{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:
ΔG (constant T, P) 
Change 
ΔG < 0 (ΔG negative) 
spontaneous 
ΔG = 0 
equilibrium (no net change) 
ΔG > 0 (ΔG positive) 
nonspontaneous 