Question

Draw an energy diagram for a one-step reaction with \(K_{\mathrm{eq}}<1 .\) Label the parts of the diagram corresponding to reactants, products, transition state, \(\Delta G^{\circ},\) and \(\Delta G^{\mp}\). Is \(\Delta G^{\circ}\) positive or negative?

Short answer

In the energy diagram for \(K_{\mathrm{eq}} < 1\), \(\Delta G^{\circ}\) is positive, showing an endergonic reaction.

Step-by-step solution

Understanding the Problem

We need to draw an energy diagram for a one-step reaction where the equilibrium constant \(K_{\mathrm{eq}} < 1\). This indicates that the products are higher in energy than the reactants because the reaction is reactant-favored.

Draw the Energy Diagram

On a graph with energy on the vertical axis and the reaction coordinate on the horizontal axis, plot the energy of the reactants as a starting point, and the products at a higher energy level to indicate that \(K_{\mathrm{eq}} < 1\).

Label Reactants and Products

Label the starting lower energy level as "Reactants" and the higher energy level where the diagram ends as "Products." This indicates the unfavorable reaction direction.

Identify and Label the Transition State

The highest point between reactants and products is the transition state. Mark this peak as "Transition State." It denotes the energy barrier that must be overcome for the reaction to proceed.

Calculate and Label \(\Delta G^{\circ}\)

The difference in energy between reactants and products on the diagram represents \(\Delta G^{\circ}\). Since products are higher in energy, \(\Delta G^{\circ}\) is positive, indicating the reaction is endergonic.

Calculate and Label \(\Delta G^{\mp}\)

The difference between the energy of the transition state and the reactants is \(\Delta G^{\mp}\), which is always positive, representing the activation energy for the reaction.

Key concepts

One-Step Reaction

A one-step reaction involves a direct conversion of reactants to products without any intermediate stages. This is characterized by a single transition state in the energy diagram. The simplicity of one-step reactions makes them ideal for illustrating basic concepts in chemical kinetics. You’ll see that the energy diagram has just one peak, which marks the transition state.

This reaction's path is straightforward: starting from the reactants, climbing up to the transition state, and descending to the products.

• Reactants are found at a lower energy level.
• The transition state marks the peak of the energy barrier.
• Products sit at a higher energy level when the equilibrium constant, \(K_{\text{eq}}<1\), indicating the reaction is not spontaneous and energetically unfavorable.

Equilibrium Constant

The equilibrium constant, \(K_{\text{eq}}\), provides a measure of the ratio of products to reactants at equilibrium. In this exercise, \(K_{\text{eq}}<1\) means that the reaction favors the reactants.

This translates on an energy diagram by showing products higher in energy than reactants.
The value of \(K_{\text{eq}}\) is crucial in predicting the reaction's direction:

• If \(K_{\text{eq}} > 1\), products are favored, often indicating an exergonic reaction.
• If \(K_{\text{eq}} < 1\), reactants are favored, suggesting the reaction does not proceed significantly towards products because the products have higher energy.

Activation Energy

Activation energy, represented as \(\Delta G^{\ddagger}\), is the energy required to reach the transition state from the reactants. It acts as a barrier to the progress of a chemical reaction.

On an energy diagram, it is depicted as the height of the peak from the level of the reactants to the transition state.
This energy must be overcome for the reactants to transform into products. Regardless of the equilibrium constant, \(\Delta G^{\ddagger}\) is always a positive value, showing that energy input is needed to surpass this hurdle.

• Higher activation energy means a slower reaction.
• Lower activation energy typically results in a faster reaction.

Endergonic Reaction

An endergonic reaction is one where the change in Gibbs free energy, \(\Delta G^{\circ}\), is positive. In the case of \(K_{\text{eq}} < 1\), the reaction is endergonic, which signifies that the energy of the products is greater than that of the reactants.

This difference in energy is reflected on the energy diagram, where the level of products is higher compared to that of the reactants. Let's break this down:

• A positive \(\Delta G^{\circ}\) means the reaction absorbs energy from its surroundings.
• Because the products are at a higher energy state, endergonic reactions are non-spontaneous and require energy input to proceed.
• This is different from exergonic reactions, which release energy and have a negative \(\Delta G^{\circ}\).