Gibbs free energy (\(\Delta G\)) is a fundamental concept in chemistry and physics. It quantifies the maximum amount of non-expansion work a system can perform at a constant temperature and pressure. Calculating \(\Delta G\) provides a thermodynamic prediction of reaction feasibility, indicating whether a chemical or physical process will occur spontaneously. A spontaneous reaction proceeds without continuous external energy input, moving the system toward a lower energy state.
The Fundamental Relationship: Using Enthalpy and Entropy
The Gibbs free energy change (\(\Delta G\)) combines two primary thermodynamic properties: enthalpy and entropy. This relationship is expressed by the defining equation: \(\Delta G = \Delta H – T\Delta S\). Here, \(\Delta H\) is the change in enthalpy, \(T\) is the absolute temperature, and \(\Delta S\) is the change in entropy. Enthalpy (\(\Delta H\)) measures the heat content change, indicating if a reaction releases heat (exothermic, negative \(\Delta H\)) or absorbs heat (endothermic, positive \(\Delta H\)). Entropy (\(\Delta S\)) quantifies the system’s disorder, where an increase in disorder is thermodynamically favorable.
The absolute temperature (\(T\)) must be measured in Kelvin. Since \(T\) is multiplied by the entropy term, a higher temperature amplifies the effect of disorder, making reactions that increase entropy more likely to be spontaneous. Unit consistency is important during calculations. Enthalpy (\(\Delta H\)) is typically reported in kilojoules (kJ), while entropy (\(\Delta S\)) is often given in joules per Kelvin (J/K). Therefore, \(\Delta S\) must frequently be converted from J/K to kJ/K so the \(T\Delta S\) term can be correctly subtracted from \(\Delta H\).
For example, if a reaction has \(\Delta H = -100 \text{ kJ/mol}\) and \(\Delta S = 150 \text{ J/(mol}\cdot\text{K)}\) at \(300 \text{ K}\), \(\Delta S\) is first converted to \(0.150 \text{ kJ/(mol}\cdot\text{K)}\). The \(\Delta G\) calculation is \(\Delta G = (-100 \text{ kJ/mol}) – (300 \text{ K} \cdot 0.150 \text{ kJ/(mol}\cdot\text{K)})\), yielding \(\Delta G = -145 \text{ kJ/mol}\). This result balances the heat released by the reaction and the increase in molecular disorder. The negative sign indicates the reaction is thermodynamically favorable under these conditions.
Calculating Standard Free Energy Change
A second method calculates the standard free energy change (\(\Delta G^{\circ}\)) using tabulated data for the free energy of formation. This method focuses on standard conditions, allowing for consistent comparison between different reactions. Standard conditions are defined by a temperature of \(298 \text{ K}\) (\(25^{\circ}\text{C}\)), a pressure of \(1 \text{ atm}\) (or \(1 \text{ bar}\)), and concentrations of \(1 \text{ M}\) for all dissolved species.
The calculation employs a summation formula analogous to Hess’s Law: \(\Delta G^{\circ} = \sum n \Delta G^{\circ}_{f} (\text{products}) – \sum m \Delta G^{\circ}_{f} (\text{reactants})\). Here, \(\Delta G^{\circ}_{f}\) is the standard free energy of formation, which is the change in free energy when one mole of a substance is formed from its constituent elements in their most stable states. The coefficients \(n\) and \(m\) are the stoichiometric coefficients from the balanced chemical equation, used as multipliers for the tabulated formation values.
To apply this method, one must look up the \(\Delta G^{\circ}_{f}\) values for every reactant and product. Elements in their standard states, such as \(\text{O}_2\) or graphite, are assigned a \(\Delta G^{\circ}_{f}\) value of zero. The calculation involves summing the multiplied formation values for all products and then subtracting the corresponding sum for all reactants. This approach bypasses the need for separate \(\Delta H\) and \(\Delta S\) calculations, relying entirely on compiled thermodynamic data to determine standard reaction favorability.
Calculating Free Energy Under Non-Standard Conditions
Most chemical reactions do not occur under the strict requirements of standard conditions, where all concentrations are precisely \(1 \text{ M}\). To determine the actual Gibbs free energy change (\(\Delta G\)), the standard value (\(\Delta G^{\circ}\)) must be adjusted for the specific, non-standard concentrations present. This adjustment uses the reaction quotient (\(Q\)) and is described by the equation: \(\Delta G = \Delta G^{\circ} + RT \ln Q\).
In this formula, \(R\) is the universal gas constant (\(8.314 \text{ J/(mol}\cdot\text{K)}\)), and \(T\) is the absolute temperature in Kelvin. The term \(\ln Q\) is the natural logarithm of the reaction quotient. \(Q\) is calculated using the instantaneous concentrations or partial pressures of reactants and products raised to their stoichiometric powers. The magnitude of \(Q\) indicates the relative amounts of products to reactants in the mixture.
If the reaction quotient \(Q\) is less than \(1\), reactants are dominant, resulting in a negative \(RT \ln Q\) value. This negative adjustment makes the overall \(\Delta G\) more negative than \(\Delta G^{\circ}\), suggesting the reaction is more favorable toward forming products. Conversely, if \(Q\) is greater than \(1\), products are dominant, the \(RT \ln Q\) term is positive, and the reaction becomes less favorable or non-spontaneous in the forward direction.
The standard free energy change (\(\Delta G^{\circ}\)) is directly connected to the equilibrium constant (\(K\)) by the equation: \(\Delta G^{\circ} = -RT \ln K\). At equilibrium, the system is balanced, and the actual free energy change (\(\Delta G\)) is zero. At this point, the reaction quotient \(Q\) equals the equilibrium constant \(K\). This relationship allows for the direct calculation of a reaction’s equilibrium constant from its thermodynamic properties, or vice versa.
Interpreting the Value: Spontaneity and Equilibrium
The calculated numerical value of the Gibbs free energy change (\(\Delta G\)) directly indicates the reaction’s thermodynamic favorability. The sign of the value provides the most immediate and useful information about the process.
A negative value for \(\Delta G\) (\(\Delta G < 0[/latex]) signifies that the reaction is spontaneous in the forward direction. This means the reaction will proceed naturally, releasing free energy. A reaction with a positive [latex]\Delta G[/latex] ([latex]\Delta G > 0\)) is non-spontaneous in the forward direction. Such a process requires a continuous input of energy, meaning the reverse reaction is thermodynamically favored.
If the calculated \(\Delta G\) is exactly zero (\(\Delta G = 0\)), the reaction system is at chemical equilibrium. At equilibrium, the rates of the forward and reverse reactions are equal, resulting in no net change in concentrations. Temperature influences the sign of \(\Delta G\), as shown in the equation \(\Delta G = \Delta H – T\Delta S\). By changing the temperature, it is possible to switch a reaction from non-spontaneous to spontaneous, controlling the feasibility of the process.