Gibbs Free Energy
Driving Forces and Gibbs Free Energy
Some reactions are spontaneous because they give off energy in the form of heat (
What happens when one of the potential driving forces behind a chemical reaction is favorable and the other is not? We can answer this question by defining a new quantity known as the Gibbs free energy (G) of the system, which reflects the balance between these forces.
The Gibbs free energy of a system at any moment in time is defined as the enthalpy of the system minus the product of the temperature times the entropy of the system.
G = H - TS
The Gibbs free energy of the system is a state function because it is defined in terms of thermodynamic properties that are state functions. The change in the Gibbs free energy of the system that occurs during a reaction is therefore equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system.
If the reaction is run at constant temperature, this equation can be written as follows.
The change in the free energy of a system that occurs during a reaction can be measured under any set of conditions. If the data are collected under standard-state conditions, the result is the standard-state free energy of reaction (
The beauty of the equation defining the free energy of a system is its ability to determine the relative importance of the enthalpy and entropy terms as driving forces behind a particular reaction. The change in the free energy of the system that occurs during a reaction measures the balance between the two driving forces that determine whether a reaction is spontaneous. As we have seen, the enthalpy and entropy terms have different sign conventions.
The entropy term is therefore subtracted from the enthalpy term when calculating
Because of the way the free energy of the system is defined,
Favorable, or spontaneous reactions: |
Conversely,
Unfavorable, or non-spontaneous reactions: |
Reactions are classified as either exothermic (
When a reaction is favored by both enthalpy (
The Effect of Temperature on the Free Energy of a Reaction
The balance between the contributions from the enthalpy and entropy terms to the free energy of a reaction depends on the temperature at which the reaction is run.
The equation used to define free energy suggests that the entropy term will become more important as the temperature increases.
Go =Ho - TSo
Since the entropy term is unfavorable, the reaction should become less favorable as the temperature increases.
Standard-State Free Energies of Reaction
Interpreting Standard-State Free Energy of Reaction Data
We are now ready to ask the obvious question: What does the value of
By definition, the value of
The sign of
Assume, for example, that we start with the following reaction under standard-state conditions, as shown in the figure below.
N2(g) + 3 H2(g)
The value of
As the reaction gradually shifts to the right, converting N2 and H2 into NH3, the value of
The Relationship Between Free Energy and Equilibrium Constants
When a reaction leaves the standard state because of a change in the ratio of the concentrations of the products to the reactants, we have to describe the system in terms of non-standard-state free energies of reaction. The difference between
The figure below shows the relationship between
N2(g) + 3 H2(g)
Data on the left side of this figure correspond to relatively small values of Qp. They therefore describe systems in which there is far more reactant than product. The sign of
Data on the far right side of this figure describe systems in which there is more product than reactant. The sign of
The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to 1. This point therefore describes the standard-state conditions, and the value of
The point at which the straight line crosses the horizontal axis describes a system for which
When Qp = Kp: G = 0
The relationship between the free energy of reaction at any moment in time (
In this equation, R is the ideal gas constant in units of J/mol-K, T is the temperature in kelvin, ln represents a logarithm to the base e, and Q is the reaction quotient at that moment in time.
As we have seen, the driving force behind a chemical reaction is zero (
0 =
We can therefore solve this equation for the relationship between
This equation allows us to calculate the equilibrium constant for any reaction from the standard-state free energy of reaction, or vice versa.
The key to understanding the relationship between
Values of
The equilibrium constant for a reaction can be expressed in two ways: Kc and Kp. We can write equilibrium constant expressions in terms of the partial pressures of the reactants and products, or in terms of their concentrations in units of moles per liter.
For gas-phase reactions the equilibrium constant obtained from
The Temperature Dependence of Equilibrium Constants
Equilibrium constants are not strictly constant because they change with temperature. We are now ready to understand why.
The standard-state free energy of reaction is a measure of how far the standard-state is from equilibrium.
But the magnitude of
As a result, the equilibrium constant must depend on the temperature of the reaction.
A good example of this phenomenon is the reaction in which NO2 dimerizes to form N2O4.
2 NO2(g)
This reaction is favored by enthalpy because it forms a new bond, which makes the system more stable. The reaction is not favored by entropy because it leads to a decrease in the disorder of the system.
NO2 is a brown gas and N2O4 is colorless. We can therefore monitor the extent to which NO2 dimerizes to form N2O4 by examining the intensity of the brown color in a sealed tube of this gas. What should happen to the equilibrium between NO2 and N2O4 as the temperature is lowered?
For the sake of argument, let's assume that there is no significant change in either
As the tube is cooled, and the entropy term becomes less important, the net effect is a shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when a sealed tube containing NO2 gas is immersed in liquid nitrogen. There is a drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC.
The Relationship Between Free Energy and Cell Potentials
The value of
The potential of an electrochemical cell is a measure of how far an oxidation-reduction reaction is from equilibrium. The Nernst equation describes the relationship between the cell potential at any moment in time and the standard-state cell potential.
Let's rearrange this equation as follows.
nFE = nFEo - RT ln Q
We can now compare it with the equation used to describe the relationship between the free energy of reaction at any moment in time and the standard-state free energy of reaction.
These equations are similar because the Nernst equation is a special case of the more general free energy relationship. We can convert one of these equations to the other by taking advantage of the following relationships between the free energy of a reaction and the cell potential of the reaction when it is run as an electrochemical cell.