# 20.4: Free Energy

The steam engine pictured below is slowly going out of style, but is still a picturesque part of the modern railroad. The water in a boiler is heated by a fire (usually fueled by coal) and turned to steam. This steam then pushes the pistons that drive the wheels of the train. It is the pressure created by the steam which allows work to be done in moving the train.

## Free Energy

Many chemical reactions and physical processes release energy that can be used to do other things. When the fuel in a car is burned, some of the released energy is used to power the vehicle. Free energy is energy that is available to do work. Spontaneous reactions release free energy as they proceed. Recall that the determining factors for spontaneity of a reaction are the enthalpy and entropy changes that occur for the system. The free energy change of a reaction is a mathematical combination of the enthalpy change and the entropy change.

$\Delta G^\text{o} = \Delta H^\text{o} - T \Delta S^\text{o}$

The symbol for free energy is $$G$$, in honor of American scientist Josiah Gibbs (1839 - 1903), who made many contributions to thermodynamics. The change in Gibbs free energy is equal to the change in enthalpy minus the mathematical product of the change in entropy multiplied by the Kelvin temperature. Each thermodynamic quantity in the equation is for substances in their standard states. The usual units for $$\Delta H$$ is $$\text{kJ/mol}$$, while $$\Delta S$$ is often reported in $$\text{J/K} \cdot \text{mol}$$. It is necessary to change the units for $$\Delta S$$ to $$\text{kJ/K} \cdot \text{mol}$$, so that the calculation of $$\Delta G$$ is in $$\text{kJ/mol}$$.

A spontaneous reaction is one that releases free energy, and so the sign of $$\Delta G$$ must be negative. Since $$\Delta H$$ and $$\Delta S$$ can be either positive or negative, depending on the characteristics of the particular reaction, there are four different general outcomes for $$\Delta G$$ and these are outlined in the table below.

 Table 20.4.1: Enthalpy, Entropy, and Free Energy Changes $$\Delta H$$ $$\Delta S$$ $$\Delta G$$ - value (exothermic) + value (disordering) always negative + value (endothermic) + value (disordering) negative at higher temperatures - value (exothermic) - value (ordering) negative at lower temperatures + value (endothermic) - value (ordering) never negative

Keep in mind that the temperature in the Gibbs free energy equation is the Kelvin temperature and so can only be positive. When $$\Delta H$$ is negative and $$\Delta S$$ is positive, the sign of $$\Delta G$$ will always be negative, and the reaction will be spontaneous at all temperatures. This corresponds to both driving forces being in favor of product formation. When $$\Delta H$$ is positive and $$\Delta S$$ is negative, the sign of $$\Delta G$$ will always be positive, and the reaction can never be spontaneous. This corresponds to both driving forces working against product formation.

When one driving force favors the reaction, but the other does not, it is the temperature that determines the sign of $$\Delta G$$. Consider first an endothermic reaction (positive $$\Delta H$$) that also displays an increase in entropy (positive $$\Delta S$$). It is the entropy term that favors the reaction. Therefore, as the temperature increases, the $$T \Delta S$$ term in the Gibbs free energy equation will begin to predominate and $$\Delta G$$ will become negative. A common example of a process which falls into this category is the melting of ice. At a relatively low temperature (below $$273 \: \text{K}$$), the melting is not spontaneous because the positive $$\Delta H$$ term "outweighs" the $$T \Delta S$$ term. When the temperature rises above $$273 \: \text{K}$$, the process becomes spontaneous because the larger $$T$$ value has tipped the sign of $$\Delta G$$ over to being negative.

When the reaction is exothermic (negative $$\Delta H$$) but undergoes a decrease in entropy (negative $$\Delta S$$), it is the enthalpy term that favors the reaction. In this case, a spontaneous reaction is dependent upon the $$T \Delta S$$ term being small relative to the $$\Delta H$$ term, so that $$\Delta G$$ is negative. The freezing of water is an example of this type of process. It is spontaneous only at a relatively low temperature. Above $$273 \: \text{K}$$, the larger $$T \Delta S$$ value causes the sign of $$\Delta G$$ to be positive, and freezing does not occur.

## Summary

• Free energy is defined.
• Relationships between enthalpy, entropy, and free energy are described.

## Contributors

• CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.