## Gibbs Free Energy or Free Enthalpy

What is Gibbs free energy? The Gibbs free energy is also known as the Gibbs function. Gibbs energy, or free enthalpy, is a thermodynamic potential that is used to calculate the maximum amount of work done in any provided thermodynamic process when the temperature and pressure of the system are held constant. The letter \[G\] represents Gibbs's free energy of any system. Due to the fact that it is also a kind of energy, its value is often expressed in either Joules or Kilojoules. The Gibbs free energy is equal to the maximum amount of work that a system is capable of doing in its environment, while simultaneously undergoing a spontaneous change or reaction (at constant temperature and pressure).

Gibbs energy is a thermodynamic property discovered by the American scientist Josiah Willard Gibbs in 1876. At that time, Gibbs was conducting experiments to determine whether or not a process could take place simultaneously and spontaneously at a given temperature. Gibbs came up with the concept for Gibbs energy while trying to predict the behaviour of systems when they were combined together. The free energy calculated by Gibbs was sometimes referred to as the available energy.

## Gibbs Free Energy Equation

The Gibbs free energy is calculated by the difference in enthalpy between the system and the product of the entropy and temperature. The equation may be written as follows:

G=H-TS

where

G= Gibbs free energy

H= Enthalpy of the system

T= Temperature of the system

S= Entropy of the system

As Gibbs free energy is a state function, it does not depend on the path of the process (i.e., it is a path independent entity).

\[\Delta G=\Delta H-\Delta \left( TS \right)\]

If the system or reaction is isothermal in nature, assuming constant temperature throughout the reaction, \[\Delta T=0\], then the following is true:

\[\Delta G=\Delta H-\left( T\Delta S \right)\]

The Gibbs Helmholtz equation is the name given to this particular equation.

\[\Delta G\,\,>\,0\]; the reaction is non-spontaneous and endergonic.

\[\Delta G\,\,<\,0\]; the reaction is spontaneous and exergonic.

\[\Delta G\,\,=\,0\]; the reaction is at equilibrium.

## What is the Equilibrium Constant?

The value of reaction quotient of a chemical reaction at chemical equilibrium is known as the equilibrium constant. Chemical equilibrium is a state that is approached by a dynamic chemical system after sufficient time has elapsed. At this point, the system's composition has no measurable tendency towards further change.

The equilibrium constant does not depend on the initial analytical concentrations of the reactant or product species in the mixture, for any particular set of reactions. Therefore, the composition of a system at equilibrium may be determined by using the initial composition of the system in conjunction with the known values of the equilibrium constants, provided that the initial composition of the system is known.

On the other hand, the value of the equilibrium constant may be affected by various reaction factors such as temperature, solvent, and ionic strength.

The thermodynamically accurate expression for the constant of equilibrium connects the activities of all of the species that are involved in the reaction. Although the idea of activity is too advanced for the typical general chemistry course, to simplify experimental measurements, the equilibrium constant of concentration approximates the activities of solutes and gases in dilute solutions with their respective molarities. However, the molarities of solids, pure liquids, and solvents cannot be used as a substitute for estimating their activities. On the other hand, the value of these actions is defined to be equal to 1.

The expression for the equilibrium constant is written as K in terms of activity for the hypothetical reactions given below:

\[bB+cC\rightleftharpoons dD+eE\]

\[{K_{eq}}={K_C}=\dfrac {{\left [D\right]^d}~{\left [E\right]^e}}{{\left [B\right]^b}~\left [C\right]^c}\]

The equilibrium constant formula generally represents the concentration and pressure of species at equilibrium.

## Standard Gibbs Free Energy

It is the change in Gibbs free energy that occurs during the formation of one mole of a compound from its component elements, when those elements are in their standard states at the most stable form of the element, that is, at \[{{\Delta }_{f}}G\].

As there is no change occurring, the standard Gibbs free energy change of formation for all elements in their standard states equals zero.

\[{\Delta_f}G={\Delta_f}{G^o}+R~T~ln{Q_f}\]

Where \[{Q_f}\] stands for the response quotient.

## Conditions for Standard State of the System

All gases involved in the process have a partial pressure of 100 kpa.

1 M is the concentration of all aqueous solutions.

The measurements are often done at a temperature of \[ \,C\,\,\left( 298K \right)\].

Pure liquids have a total (hydrostatic) pressure of 1 atm.

For the pure solid at 1 atm pressure.

In equilibrium, \[{{\Delta }_{f}}G\] is equal to zero, and Qf is to K; therefore, the equation becomes:

\[{{\Delta }_{f}}G=-R~T ~ln K\]

where $K$ is the equilibrium constant.

Consider a spontaneous chemical process in a closed system at constant temperature and pressure, without non-PV work. The Clausius inequality $\Delta S > \dfrac{Q}{T_{surr}}$ for such a system converts into a condition for the change in Gibbs free energy.

$\Delta G < 0$

## Relation Between Gibbs Free Energy and Equilibrium Constant

The standard free energy change of the reaction, \[\Delta {G^o}\](which is equal to the difference in the free energies of formation of the products and reactants both in their standard states), is connected to the reaction's free energy change in any state, (where equilibrium has not been achieved).

\[\Delta G=\Delta G{}^\circ+RT~ln Q\]

Here, Q is the reaction quotient.

At equilibrium,

\[~\Delta G\,=\,0\], and Q becomes equal to the equilibrium constant, K at equilibrium. Hence, the equation will become:

\[\Delta G{}^\circ =2.303RT\log {{K}_{eq}}\]

Here, R is equal to 8.314 J mol-1 k-1 or \[\,0.008314~kJ~mo{{l}^{-1}}~{{k}^{-1}}\].

T corresponds to the temperature on the Kelvin scale.

The relation between the equilibrium constant and Gibbs free energy can be derived using the following equation:

\[{K_{eq}}={K_C}=\dfrac {{\left [D\right]^d}~{\left [E\right]^e}}{{\left [B\right]^b}~\left [C\right]^c}\]

## Relationship between Gibbs Free Energy and EMF of a Cell

The Gibbs energy change \[~\Delta G\] of galvanic cells is proportional to the amount of electrical work the cell does.

\[\Delta G=-nF{{E}_{cell}}\]

where \[n\,\,=\] no. of moles of electron involved

\[F\,\,=\] constant

\[E\,\,=\] EMF of the cell

\[F\,\,=\,\,1\,{\text{Faraday}}\,\,=\,\,96500\,\,\text{coulombs}\]

If reactants and products are in their standard states,

\[\Delta G{}^\circ =nFE{{{}^\circ }_{cell}}\]

\[E{{{}^\circ }_{cell}}\] and equilibrium constant, \[{{K}_{eq}}\]

\[\Delta G=\Delta G{}^\circ +RT\ln Q\]

\[0=\Delta G{}^\circ +RT \ln\text{ Keq}\]

\[ln\text{ Keq}=-\frac{\Delta G{}^\circ }{RT}\]

\[{{K}_{eq}}={{e}^{-\,\frac{\Delta G{}^\circ }{RT}}}\]

\[\Delta \mathbf{G}{}^\circ \] and Equilibrium Condition of Spontaneous, Non-Spontaneous and Equilibrium

Table 1: Conditions for Equilibrium for Various Processes

## Conclusion

Gibbs energy is a state function, which is related to entropy, temperature and enthalpy changes of the system, and measure of its capacity to do useful work. The decrease in Gibbs free energy during a process gives the maximum possible useful work that can be obtained from the system. The free energy of a system is a measure of its capacity to do useful work. When \[\Delta G\] is negative, the given reaction is spontaneous. For a reversible reaction, \[\Delta G\] is negative for the forward as well as backward reaction. \[\Delta G\] is zero at equilibrium and there is no further free energy change.

## FAQs on Standard Gibbs Free Energy and Equilibrium Constant for JEE

1. Is Gibbs's free energy an example of a state function? Explain.

Gibbs's free energy is an example of a state function because it is explained by other state functions. Gibbs free energy is defined by other state functions (G=H-TS), enthalpy, and entropy. Additionally, Gibbs free energy is a state property because it is defined by final-initial state and is not dependent on the path it undergoes during the change. Gibbs free energy is also an extensive property.

2. What is the relationship between K_{P} and K_{C}?

The equilibrium constants for a perfect gaseous mixture are K_{P} and K_{C}. K_{P} is the appropriate equilibrium constant to use when expressing equilibrium concentrations in terms of atmospheric pressure, while K_{C} is the appropriate equilibrium constant to use when expressing equilibrium concentrations in terms of molarity. For a general equilibrium reaction,

$\mathrm{aA}+\mathrm{bB} \rightleftharpoons \mathrm{cC}+\mathrm{dD}$

The connection between K_{p} and K_{c} may be expressed as follows:

$K_{p}=K_{c}(R T)^{\Delta n g}$

Here, $\Delta n g$= no. of moles of gaseous products - no. of moles of gaseous reactants for balanced chemical equations.