Entropy and Free Energy by Owen Borville 10.27.2025
Spontaneous processes: A process that does occur under a specific set of conditions is called a spontaneous process. A process that does not occur under a specific set of conditions is called nonspontaneous process.
Spontaneous processes examples: rock rolling down a hill, heat flowing from a hot object to a cold object, ice melting into water at room temperature, gas expanding in a container, iron rusting, dissolution of a sugar cube in water.
Nonspontaneous processes examples: electrolysis of water, photosynthesis, charging a battery, pumping water uphill, freezing water at room temperature, reverse osmosis, formation of glucose.
A process that results in a decrease in the energy of a system often is spontaneous: CH4(g) +2O2(g) =>CO2(g) + 2H2O(l) ΔH° = -890.4 kJ/mol
The sign of ΔH alone is insufficient to predict spontaneity in every circumstance: H2O(l) => H2O(s) T>0° C; ΔH° = -6.01 kJ/mol
To predict spontaneity, both the enthalpy and entropy must be known.
Entropy (S) of a system is a measure of how spread out or how dispersed the system's energy is.
Spontaneity is favored by an increase in entropy
S = k ln W
where k is the Boltzmann constant (1.38 x 10-23 J/K)
W is the number of different arrangements.
The number of arrangements possible is given by W = X^N
X is the number of cells in a volume
N is the number of molecules
The possible arrangements with 2 molecules and only 2 cells is = 2^2 = 4
To draw all possibilities: 4^2= 16 There are three possible states for this system: (1) one molecule on each side (8 possible arrangements) (2) Both molecules on the left (four possible arrangements) (3) Both molecules on the right (four possible arrangements) The most probable state has the largest number of arrangements.
Entropy Changes in a System: Calculating ΔSsys The change in entropy for a system is the difference in entropy of the final state and the entropy of the initial state.
ΔSsys= Sfinal - Sinitial
Alternatively, using S=klnW and volume: ΔSsys = nRln(Vfinal/Vinitial)
Remember that for a process to be spontaneous, something must favor spontaneity. If the process is spontaneous but not exothermic (in this case, there is no enthalpy change), then we should expect ΔSsys to be positive.
Standard Entropy S° is the absolute entropy of a substance at 1 atm. Temperature is not part of the standard state definition and must be specified.
Trends in Entropy S°
S° liquid > S° solid
S° gas > S° liquid
S° increases with molar mass
S° increases with molecular complexity
S° increases with the mobility of a phase (for an element with two or more allotropes)
In addition to translational motion (without rotation), molecules exhibit vibrations and rotations.
For a chemical reaction: aA + bB => cC + dD
ΔS°rxn = [cS°(C) + dS°(D)]-[aS°(A) + bS°(B)]
Alternatively, ΔS°rxn = ∑nS°(products)-∑mS°(reactants)
Remember to multiply each standard entropy value by the correct stoichiometric coefficient from a balanced chemical equation.
Qualitatively Predicting the Sign of ΔS°sys: Several processes that lead to an increase in entropy are: melting, vaporization or sublimation, temperature increase, reaction resulting in a greater number of gas molecules.
The process of dissolving a substance can lead to either an increase or a decrease in entropy, depending on the nature of the solute: Molecular solutes (ex. sugar): entropy increases. Ionic compounds: entropy could decrease or increase.
Entropy Changes: Correctly predicting the spontaneity of a process requires consideration of entropy changes in both the system and the surroundings.
For example, if an ice cube spontaneously melts in a room at 25° degrees C, the system, ice, has a positive ΔS, but the surroundings and everything else have a negative ΔS.
If a cup of hot water spontaneously cools to room temperature, the system, hot water, has a negative ΔS, but the surroundings and everything else have a positive ΔS.
The entropy of both the system AND surroundings are important.
The change in entropy of the surroundings is directly proportional to the enthalpy of the system: ΔSsurr = (-ΔHsys)/T
The second law of thermodynamics states that for a process to be spontaneous, ΔS(universe) must be positive.
ΔS(universe) = ΔSsys + ΔSsurr
ΔS(universe) > 0 for a spontaneous process
ΔS(universe) < 0 for a nonspontaneous process
ΔS(universe) = 0 for an equilibrium process
When calculating enthalpies, remember to use the correct units with joules and kJ when combining terms.
Third law of thermodynamics states that the entropy of a perfect crystalline substance is zero at absolute zero. Entropy increases in a substance as temperature increases from absolute zero.
Measurements on the surroundings are seldom made, limiting the use of the second law of thermodynamics.
Gibbs Free Energy (G) (also called Free Energy) can be used to express spontaneity more directly.
G = H-TS (Free Energy = enthalpy of heat - temperature*entropy)
The change in free energy for a system is: ΔG = ΔH-TΔS
ΔG = Change in Gibbs Free Energy
ΔH = Change in enthalpy of heat
T = Temperature in Kelvin
ΔS = Change in entropy
Using Gibbs Free Energy, it is possible to make predictions on spontaneity:
ΔG = ΔH-TΔS
ΔG < 0 The reaction is spontaneous in the forward direction.
ΔG > 0 The reaction is nonspontaneous in the forward direction.
ΔG = 0 The system is at equilibrium.
Standard Free Energy of Reaction (ΔG°rxn) is free-energy change for a reaction when it occurs under standard-state conditions.
The following conditions define the standard states of pure substances and solutions:
Gases = 1 atm pressure
Liquids = pure liquid
Solids = pure solid
Elements = the most stable allotropic form at 1 atm and 25°C
Solutions = 1 molar concentration
For a chemical reaction aA + bB => cC + dD
ΔG°rxn = [cΔG°f(C) + dΔG°f(D)]-[aΔG°f(A) + bΔG°f(B)]
Alternatively, ΔG°rxn = ∑nΔG°f (products)-∑mΔG°f (reactants)
ΔG°f for any element in its most stable allotropic form at 1 atm is defined as zero.
Many biological reactions have a positive ΔG° value, making the reaction nonspontaneous.
Nonspontaneous reactions can be coupled with spontaneous reactions in order to drive a process forward:
alanine + glycine => alanylglycine ΔG° = 29 kJ/mol
ATP + H2O => ADP + H3PO4 ΔG° = -31 kJ/mol
---------------------------------------------------------------
ATP + H2O + alanine + glycine => ADP + H3PO4 + alanylglycene
ΔG° = 29 kJ/mol + -31 kJ/mol = -2 kJ/mol
This combination of spontaneous and nonspontaneous reactions is called energy coupling and is essential for all living organisms to power processes that would not naturally occur on their own.
Energy coupling is the process where an exergonic (energy-releasing) reaction drives an endergonic (energy-requiring) reaction. In biological systems, this is most often achieved by using the energy released from the breakdown of ATP (adenosine triphosphate) to power other cellular processes that would otherwise not happen spontaneously.
Spontaneous processes: A process that does occur under a specific set of conditions is called a spontaneous process. A process that does not occur under a specific set of conditions is called nonspontaneous process.
Spontaneous processes examples: rock rolling down a hill, heat flowing from a hot object to a cold object, ice melting into water at room temperature, gas expanding in a container, iron rusting, dissolution of a sugar cube in water.
Nonspontaneous processes examples: electrolysis of water, photosynthesis, charging a battery, pumping water uphill, freezing water at room temperature, reverse osmosis, formation of glucose.
A process that results in a decrease in the energy of a system often is spontaneous: CH4(g) +2O2(g) =>CO2(g) + 2H2O(l) ΔH° = -890.4 kJ/mol
The sign of ΔH alone is insufficient to predict spontaneity in every circumstance: H2O(l) => H2O(s) T>0° C; ΔH° = -6.01 kJ/mol
To predict spontaneity, both the enthalpy and entropy must be known.
Entropy (S) of a system is a measure of how spread out or how dispersed the system's energy is.
Spontaneity is favored by an increase in entropy
S = k ln W
where k is the Boltzmann constant (1.38 x 10-23 J/K)
W is the number of different arrangements.
The number of arrangements possible is given by W = X^N
X is the number of cells in a volume
N is the number of molecules
The possible arrangements with 2 molecules and only 2 cells is = 2^2 = 4
To draw all possibilities: 4^2= 16 There are three possible states for this system: (1) one molecule on each side (8 possible arrangements) (2) Both molecules on the left (four possible arrangements) (3) Both molecules on the right (four possible arrangements) The most probable state has the largest number of arrangements.
Entropy Changes in a System: Calculating ΔSsys The change in entropy for a system is the difference in entropy of the final state and the entropy of the initial state.
ΔSsys= Sfinal - Sinitial
Alternatively, using S=klnW and volume: ΔSsys = nRln(Vfinal/Vinitial)
Remember that for a process to be spontaneous, something must favor spontaneity. If the process is spontaneous but not exothermic (in this case, there is no enthalpy change), then we should expect ΔSsys to be positive.
Standard Entropy S° is the absolute entropy of a substance at 1 atm. Temperature is not part of the standard state definition and must be specified.
Trends in Entropy S°
S° liquid > S° solid
S° gas > S° liquid
S° increases with molar mass
S° increases with molecular complexity
S° increases with the mobility of a phase (for an element with two or more allotropes)
In addition to translational motion (without rotation), molecules exhibit vibrations and rotations.
For a chemical reaction: aA + bB => cC + dD
ΔS°rxn = [cS°(C) + dS°(D)]-[aS°(A) + bS°(B)]
Alternatively, ΔS°rxn = ∑nS°(products)-∑mS°(reactants)
Remember to multiply each standard entropy value by the correct stoichiometric coefficient from a balanced chemical equation.
Qualitatively Predicting the Sign of ΔS°sys: Several processes that lead to an increase in entropy are: melting, vaporization or sublimation, temperature increase, reaction resulting in a greater number of gas molecules.
The process of dissolving a substance can lead to either an increase or a decrease in entropy, depending on the nature of the solute: Molecular solutes (ex. sugar): entropy increases. Ionic compounds: entropy could decrease or increase.
Entropy Changes: Correctly predicting the spontaneity of a process requires consideration of entropy changes in both the system and the surroundings.
For example, if an ice cube spontaneously melts in a room at 25° degrees C, the system, ice, has a positive ΔS, but the surroundings and everything else have a negative ΔS.
If a cup of hot water spontaneously cools to room temperature, the system, hot water, has a negative ΔS, but the surroundings and everything else have a positive ΔS.
The entropy of both the system AND surroundings are important.
The change in entropy of the surroundings is directly proportional to the enthalpy of the system: ΔSsurr = (-ΔHsys)/T
The second law of thermodynamics states that for a process to be spontaneous, ΔS(universe) must be positive.
ΔS(universe) = ΔSsys + ΔSsurr
ΔS(universe) > 0 for a spontaneous process
ΔS(universe) < 0 for a nonspontaneous process
ΔS(universe) = 0 for an equilibrium process
When calculating enthalpies, remember to use the correct units with joules and kJ when combining terms.
Third law of thermodynamics states that the entropy of a perfect crystalline substance is zero at absolute zero. Entropy increases in a substance as temperature increases from absolute zero.
Measurements on the surroundings are seldom made, limiting the use of the second law of thermodynamics.
Gibbs Free Energy (G) (also called Free Energy) can be used to express spontaneity more directly.
G = H-TS (Free Energy = enthalpy of heat - temperature*entropy)
The change in free energy for a system is: ΔG = ΔH-TΔS
ΔG = Change in Gibbs Free Energy
ΔH = Change in enthalpy of heat
T = Temperature in Kelvin
ΔS = Change in entropy
Using Gibbs Free Energy, it is possible to make predictions on spontaneity:
ΔG = ΔH-TΔS
ΔG < 0 The reaction is spontaneous in the forward direction.
ΔG > 0 The reaction is nonspontaneous in the forward direction.
ΔG = 0 The system is at equilibrium.
Standard Free Energy of Reaction (ΔG°rxn) is free-energy change for a reaction when it occurs under standard-state conditions.
The following conditions define the standard states of pure substances and solutions:
Gases = 1 atm pressure
Liquids = pure liquid
Solids = pure solid
Elements = the most stable allotropic form at 1 atm and 25°C
Solutions = 1 molar concentration
For a chemical reaction aA + bB => cC + dD
ΔG°rxn = [cΔG°f(C) + dΔG°f(D)]-[aΔG°f(A) + bΔG°f(B)]
Alternatively, ΔG°rxn = ∑nΔG°f (products)-∑mΔG°f (reactants)
ΔG°f for any element in its most stable allotropic form at 1 atm is defined as zero.
Many biological reactions have a positive ΔG° value, making the reaction nonspontaneous.
Nonspontaneous reactions can be coupled with spontaneous reactions in order to drive a process forward:
alanine + glycine => alanylglycine ΔG° = 29 kJ/mol
ATP + H2O => ADP + H3PO4 ΔG° = -31 kJ/mol
---------------------------------------------------------------
ATP + H2O + alanine + glycine => ADP + H3PO4 + alanylglycene
ΔG° = 29 kJ/mol + -31 kJ/mol = -2 kJ/mol
This combination of spontaneous and nonspontaneous reactions is called energy coupling and is essential for all living organisms to power processes that would not naturally occur on their own.
Energy coupling is the process where an exergonic (energy-releasing) reaction drives an endergonic (energy-requiring) reaction. In biological systems, this is most often achieved by using the energy released from the breakdown of ATP (adenosine triphosphate) to power other cellular processes that would otherwise not happen spontaneously.