Question

What information can be determined from \(\Delta G\) for a reaction? Does one get the same information from \(\Delta G^{\circ},\) the standard free energy change? \(\Delta G^{\circ}\) allows determination of the equilibrium constant \(K\) for a reaction. How? How can one estimate the value of \(K\) at temperatures other than \(25^{\circ} \mathrm{C}\) for a reaction? How can one estimate the temperature where \(K=1\) for a reaction? Do all reactions have a specific temperature where \(K=1 ?\)

Short answer

The Gibbs free energy change (∆G) determines the spontaneity of a reaction, while the standard free energy change (∆G°) refers to a reaction under standard conditions. The equilibrium constant (K) can be determined from ∆G° using the equation \(\Delta G^{\circ} = -RT \ln K\). To estimate K at different temperatures, the Van't Hoff equation can be used. The temperature at which K=1 can be estimated using the ∆G° relationship, but not all reactions have a specific temperature where K=1.

Step-by-step solution

Understanding Gibbs Free Energy and Standard Free Energy Change

The Gibbs free energy change (∆G) in a reaction measures the spontaneity of a process or reaction. If ∆G is negative, the reaction is spontaneous and will proceed in the forward direction. If ∆G is positive, the reaction is non-spontaneous, and the reverse reaction is favored. When ∆G is equal to zero, the reaction is at equilibrium. The standard free energy change (∆G°) is the free energy change of a reaction under standard conditions (1 atm of pressure, 25°C temperature, and 1 M concentration). While ∆G provides information about the thermodynamics of a reaction under specific conditions, ∆G° refers to the reaction under standard conditions.

Relation between ∆G° and the equilibrium constant K

The equilibrium constant (K) is related to the standard free energy change according to the following equation: \[ \Delta G^{\circ} = -RT \ln K\] Where R is the ideal gas constant (8.314 J/molK), T is the temperature in Kelvin, and ln refers to the natural logarithm. From this equation, we can see that as ∆G° gets more negative, the equilibrium constant (K) becomes larger, meaning the reaction is more favorable and proceeds predominantly in the forward direction. Conversely, if ∆G° is positive, K will be smaller, and the reaction will be less favorable.

Estimating K at temperatures other than 25°C

To estimate the value of K at temperatures other than standard (25°C or 298K), we can use the Van't Hoff equation: \[ \frac{d\ln K}{dT} = \frac{\Delta H^{\circ}}{RT^2}\] Where ∆H° is the standard enthalpy change, which is typically given or can be determined experimentally. To find K at a specific temperature, T', we can integrate the Van't Hoff equation and solve for K(T') given K at the standard temperature of 298K.

Estimating the temperature where K=1 for a reaction

To estimate the temperature where the equilibrium constant K=1, we can use the equation relating ∆G° to K: \[ \Delta G^{\circ} = -RT \ln K\] When K=1, ln(K) = 0. Therefore, ∆G° = 0. We can then solve for the temperature (T) at which this occurs: \[ T = \frac{\Delta G^{\circ}}{-R\ln K}\]

Do all reactions have a specific temperature where K=1?

Not all reactions have a specific temperature where K=1. Some reactions, like certain endothermic reactions with positive standard enthalpy change (∆H° > 0) and significant temperature dependence, may not have a temperature where K=1. In these cases, the reaction equilibrium will always favor either the products or the reactants.