Question

List three different ways to calculate the standard free energy change, \(\Delta G^{\circ}\), for a reaction at \(25^{\circ} \mathrm{C}\). How is \(\Delta G^{\circ}\) estimated at temperatures other than \(25^{\circ} \mathrm{C}\) ? What assumptions are made?

Short answer

There are three ways to calculate ΔG° for a reaction at $25^{\circ}\mathrm{C}$: 1) Using the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°) with the equation \( \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ \); 2) Using the equilibrium constant (K) with the equation \( \Delta G^\circ = -RT \ln K \); 3) For redox reactions, using the standard electrode potential (E°) with the equation \( \Delta G^\circ = -nFE^\circ \). To estimate ΔG° at temperatures other than $25^{\circ}\mathrm{C}$, the van't Hoff equation \( \frac{\Delta G_2^\circ - \Delta G_1^\circ}{T_2 - T_1} = -\frac{\Delta H^\circ}{R}(\frac{1}{T_2} - \frac{1}{T_1}) \) can be used. The assumptions made are: 1) Standard state conditions are considered; 2) ΔH° and ΔS° are temperature-independent or their temperature-dependence is negligible; 3) The van't Hoff equation assumes ΔH° is constant over the temperature range considered.

Step-by-step solution

1. Calculating ΔG° using the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°)

To calculate the standard free energy change (ΔG°) at 25°C using the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°), use the following equation: \( \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ \) where T is the temperature in Kelvin (25°C = 298.15 K).

2. Calculating ΔG° from the equilibrium constant (K)

The standard free energy change (ΔG°) at 25°C can also be calculated using the equilibrium constant (K) of the reaction using the following equation: \( \Delta G^\circ = -RT \ln K \) where R is the universal gas constant (8.314 J/mol·K) and T is the temperature in Kelvin (25°C = 298.15 K).

3. Calculating ΔG° using the relationship between the Gibbs free energy and the standard electrode potential (E°)

For redox reactions, the standard free energy change (ΔG°) can be calculated using the relationship between the Gibbs free energy change and the standard electrode potential (E°) using the following equation: \( \Delta G^\circ = -nFE^\circ \) where n is the number of electrons transferred in the redox reaction, F is Faraday's constant (96,485 C/mol or 96.485 kJ/mol·V), and E° is the standard electrode potential of the reaction.

Estimating ΔG° at temperatures other than 25°C

At temperatures other than 25°C, the standard free energy change (ΔG°) can be estimated using the van't Hoff equation: \( \frac{\Delta G_2^\circ - \Delta G_1^\circ}{T_2 - T_1} = -\frac{\Delta H^\circ}{R}(\frac{1}{T_2} - \frac{1}{T_1}) \) where ΔG₁° and ΔG₂° are the standard free energy changes at temperatures T₁ and T₂ (in Kelvin) respectively, R is the universal gas constant, and ΔH° is the standard enthalpy change of the reaction.

Assumptions made

The assumptions made in the calculation and estimation of the standard free energy change (ΔG°) are: 1. The standard state conditions, including a temperature of 25°C, 1 atm pressure, and 1 M concentration for each reactant and product, are considered. 2. The enthalpy and entropy changes (ΔH° and ΔS°) are temperature-independent or temperature-dependence is negligible. 3. The van't Hoff equation assumes that the standard enthalpy change (ΔH°) is constant over the temperature range considered.

Key concepts

Standard Enthalpy Change

The standard enthalpy change (ΔH°) is a measure of the heat absorbed or released during a chemical reaction under standard state conditions. It's important because it helps predict whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).

When calculating Gibbs Free Energy (\(ΔG^\) using ΔH° at 25°C, we use the formula:

• \(ΔG^\circ = ΔH^\circ - TΔS^\ \circ\)

This formula helps in understanding the balance between enthalpy and entropy changes in determining the spontaneity of a reaction.

It is assumed that ΔH° remains constant with temperature, allowing us to use it across different calculations and conditions.

Standard Entropy Change

Standard entropy change (ΔS°) measures the disorder or randomness of a system when a reaction occurs under standard state conditions. A positive ΔS° indicates increased disorder, while a negative value suggests a decreased disorder.

Entropy is crucial in calculating the Gibbs Free Energy, particularly its role in determining whether a reaction is spontaneous. The equation, \(ΔG^\circ = ΔH^\circ - T ΔS^\circ\), includes ΔS° to account for this disorder's impact. The temperature in Kelvin (T) amplifies any changes in ΔS°, making it essential to consider even small entropy changes.

Equilibrium Constant

The equilibrium constant (K) measures a reaction's position at equilibrium under standard conditions. It is a ratio of the concentration of products to reactants, each raised to the power of their respective coefficients in the balanced equation.

For calculating Gibbs Free Energy from K, the equation \(ΔG^\circ = -RT \ln K\) is used. Here, R is the universal gas constant, and T is the temperature in Kelvin. This relationship shows that reactions with large K values favor products and tend to be spontaneous, reflected by a negative ΔG° value.

Gibbs Free Energy

Gibbs Free Energy (ΔG) is a measure of a system's maximum reversible work at constant temperature and pressure. It tells us whether a reaction is spontaneous or requires energy input. Negative ΔG values indicate spontaneous reactions, while positive values suggest non-spontaneity.

In redox reactions, the ΔG° can be calculated from the standard electrode potential (E°) using the formula:

• \(ΔG^\circ = -nFE^\circ\)

Where 'n' is the number of electrons transferred, and 'F' is Faraday's constant. This equation connects electrochemical cell potentials with reaction spontaneity.

Standard Electrode Potential

Standard electrode potential (E°) reflects the voltage associated with a redox reaction under standard conditions. It provides insight into the electron transfer tendency during a reaction. A positive E° indicates that a reaction is more likely to occur spontaneously.

You can utilize the E° to find the standard free energy change of a redox reaction through the equation \(ΔG^\circ = -nFE^\circ\). This links the thermodynamic and electrochemical perspectives of reactions, showing how changes in electron flow relate to energy transformations.

Van't Hoff Equation

The Van’t Hoff equation helps estimate how equilibrium constants vary with temperature. The equation used in this context is:

• \[\frac{\Delta G_2^\circ - \Delta G_1^\circ}{T_2 - T_1} = -\frac{\Delta H^\circ}{R}(\frac{1}{T_2} - \frac{1}{T_1})\]

This provides a means of estimating ΔG° for temperatures other than 25°C. The Van’t Hoff equation assumes that ΔH° remains constant over the temperature range. This method is essential for reactions that do not strictly occur at standard temperature.

Standard State Conditions

Standard state conditions are a set of assumptions applied in thermodynamic calculations to have consistency across systems. These conditions include:

• Temperature of 25°C (298.15 K)
• Pressure of 1 atm
• Concentration of 1 M for all reactants and products

These standardized parameters provide a reference point to compare different reactions' thermodynamics. It establishes a common ground for evaluating changes in enthalpy, entropy, and free energy across various conditions. Moreover, calculations assume temperature-independent ΔH° and ΔS°, simplifying the characterization of chemical processes.