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

Three different ways to calculate the standard free energy change, \(\Delta G^{\circ}\), at 25°C are: 1. Using the Gibbs-Helmholtz equation: \(\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\) 2. Using the equilibrium constant: \(\Delta G^{\circ} = -RT \ln(K)\) 3. Using data tables and the equation: \(\Delta G^{\circ} = \sum \Delta G_f^{\circ}(\text{products}) - \sum \Delta G_f^{\circ}(\text{reactants})\) To estimate \(\Delta G^{\circ}\) at other temperatures, we can use the van't Hoff equation: \[\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\] Assumptions made while estimating \(\Delta G^{\circ}\) at a different temperature include: 1. The reaction is assumed to be at equilibrium. 2. The standard free energy change, enthalpy change, and entropy change are assumed to be constant over the temperature range being considered. 3. The van't Hoff equation assumes that the change in standard free energy and enthalpy with temperature is negligible.

Step-by-step solution

Method 1: Using Gibbs-Helmholtz equation

The standard free energy change, \(\Delta G^{\circ}\), can be calculated using the Gibbs-Helmholtz equation: \[\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\] Here, we need to first calculate the standard enthalpy change, \(\Delta H^{\circ}\), and the standard entropy change, \(\Delta S^{\circ}\) of the reaction, then plug them into this equation to calculate the standard free energy change at a given temperature, T (in Kelvin).

Method 2: Using equilibrium constant

We can also calculate \(\Delta G^{\circ}\) using the relationship between the equilibrium constant, K, and the standard free energy change: \[\Delta G^{\circ} = -RT \ln(K)\] Here, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature (in Kelvin). To use this method, we need to first determine the equilibrium constant, K, for the reaction at the desired temperature, and then plug it into this equation to calculate \(\Delta G^{\circ}\).

Method 3: Using the data tables

Another way to calculate \(\Delta G^{\circ}\) is by using the data tables that contain the standard free energy of formation, \(\Delta G_f^{\circ}\), for various substances. The standard free energy change for a reaction can be calculated using the equation: \[\Delta G^{\circ} = \sum \Delta G_f^{\circ}(\text{products}) - \sum \Delta G_f^{\circ}(\text{reactants})\] In this method, we need to look up the standard free energy of formation for each substance involved in the reaction and then use this equation to calculate \(\Delta G^{\circ}\).

Estimating \(\Delta G^{\circ}\) at other temperatures

To estimate the standard free energy change at a different temperature, we can use the following van't Hoff equation: \[\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\] Here, K1 and K2 are the equilibrium constants at the initial (T1) and final (T2) temperatures, respectively. First, we need to determine \(\Delta H^{\circ}\) and K1 for the reaction at the initial temperature using one of the methods mentioned above. Then, we can use this equation to calculate K2 at the desired temperature T2. Finally, we can use the relationship between K and \(\Delta G^{\circ}\) mentioned in Method 2 to calculate the standard free energy change at the new temperature.

Assumptions made

There are a few assumptions made while estimating \(\Delta G^{\circ}\) at a different temperature: 1. The reaction is assumed to be at equilibrium. 2. The standard free energy change, enthalpy change, and entropy change are assumed to be constant over the temperature range being considered. 3. The van't Hoff equation assumes that the change in standard free energy and enthalpy with temperature is negligible.

Key concepts

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz equation is a fundamental tool in thermodynamics that allows us to calculate the standard free energy change (\( \Delta G^{\circ} \)) for a chemical reaction. This equation is expressed as:
\[ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \]
where \( \Delta H^{\circ} \) is the standard enthalpy change, \( \Delta S^{\circ} \) is the standard entropy change, and \( T \) is the temperature in Kelvin.
This equation shows that the free energy change of a reaction is influenced by both the enthalpy and entropy changes.

• Positive \( \Delta G^{\circ} \): This indicates a non-spontaneous reaction under standard conditions.
• Negative \( \Delta G^{\circ} \): This means the reaction is spontaneous under standard conditions.

To use the Gibbs-Helmholtz equation, you need to determine \( \Delta H^{\circ} \) and \( \Delta S^{\circ} \) from experimental data or tables and substitute them into the equation along with the temperature in Kelvin. Understanding this equation helps predict whether a reaction will occur on its own or if it requires external energy input.

Equilibrium Constant

The equilibrium constant (\( K \)) is a number that gives insight into the position of balance in a reversible chemical reaction at a given temperature. It reflects the ratio of product concentrations to reactant concentrations when the system is at equilibrium:

• \( K > 1 \): Products are favored at equilibrium.
• \( K < 1 \): Reactants are favored at equilibrium.

\( \Delta G^{\circ} \) is related to the equilibrium constant through the relationship:
\[ \Delta G^{\circ} = -RT \ln(K) \]
Here, \( R \) is the ideal gas constant (8.314 J/(mol·K)), and \( T \) is the temperature in Kelvin. This equation indicates how a change in \( \Delta G^{\circ} \) affects \( K \):

• If \( \Delta G^{\circ} \) is negative, \( K \) is greater than 1, suggesting a favorable formation of products.
• If \( \Delta G^{\circ} \) is positive, \( K \) is less than 1, implying a reaction that does not favor product formation under standard conditions.

Understanding this relationship helps predict how a reaction will behave in equilibrium and the extent to which reactants are converted to products.

Free Energy of Formation

The free energy of formation, given by \( \Delta G_f^{\circ} \), is a measure of the energy change when one mole of a compound is formed from its constituent elements in their standard states.
This value provides crucial information for calculating the standard free energy change of a reaction using:
\[ \Delta G^{\circ} = \sum \Delta G_f^{\circ}(\text{products}) - \sum \Delta G_f^{\circ}(\text{reactants}) \]
By using tabulated values for \( \Delta G_f^{\circ} \), you can easily find \( \Delta G^{\circ} \) for reactions without direct experimentation.

• Spontaneity: Negative \( \Delta G^{\circ} \) indicates spontaneous reactions.
• Positive \( \Delta G^{\circ} \): Reactions are non-spontaneous and likely need energy input.

Understanding free energy of formation helps chemists and engineers design viable processes by predicting which reactions are thermodynamically favorably based solely on starting materials and products.

Temperature Dependence

The temperature dependence of a reaction determines how temperature variations influence the thermodynamic properties like free energy, entropy, and enthalpy during a reaction.
Standard free energy change (\( \Delta G^{\circ} \)) is particularly affected by temperature, as dictated by the Gibbs-Helmholtz equation.Unlike \( \Delta H^{\circ} \) and \( \Delta S^{\circ} \), which are often approximated to be constant over small temperature ranges, \( \Delta G^{\circ} \) has a more pronounced temperature dependency.

• For exothermic reactions (\( \Delta H^{\circ} < 0 \)), increasing temperature can make \( \Delta G^{\circ} \) less negative, potentially reducing spontaneity.
• For endothermic reactions (\( \Delta H^{\circ} > 0 \)), higher temperatures may make \( \Delta G^{\circ} \) more negative, increasing spontaneity.

Assessing temperature effects helps predict the feasibility of reactions under non-standard conditions, such as industrial temperatures, and guides adjustments needed to maintain desired reaction pathways.

Van't Hoff Equation

The Van't Hoff equation is pivotal in understanding how temperature affects the equilibrium constant (\( K \)) of a reaction. Given the relationship:
\[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]
this equation allows us to estimate \( K \) at a different temperature given a known \( K \) at another temperature and the standard enthalpy change \( \Delta H^{\circ} \).To use the Van't Hoff equation:

• Calculate or obtain \( \Delta H^{\circ} \) from sources or experiments.
• Use the known \( K_1 \) for an initial temperature \( T_1 \).
• Substitute \( \Delta H^{\circ} \), \( T_1 \), and the desired \( T_2 \) into the equation to find \( K_2 \).

This is crucial for predicting how changes in temperature will shift the equilibrium of a reaction, which is essential for processes like chemical manufacturing and environmental assessments.