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# Free Energy

## Defines free energy and how entropy and enthalpy affect its value.

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Free Energy

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The steam engine pictured above 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.

$\Delta G^\circ=\Delta H^\circ - T \Delta S^\circ$

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 kJ/mol, while  $\Delta S$ is often reported in J/K • mol. It is necessary to change the units for  $\Delta S$ to kJ/K • mol, so that the calculation of  $\Delta G$ is in kJ/mol.

A spontaneous reaction is one that releases free energy, and so the sign of  $\Delta G$ must be negative. Since both $\Delta H$ 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:

 $\Delta H$ $\Delta S$ $\Delta G$ − 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.

#### Summary

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

#### Practice

2. What happens to the total energy when the ball rolls down the slide?
3. How does  $H$ change in a spontaneous reaction?
4. How does  $S$ change in a spontaneous reaction?

#### Review

1. What do spontaneous reactions do?
2. What are the units for $\Delta H$ ?
3. What are the units for $\Delta S$ ?