Problem 10 - Biochemistry: Concepts and Connections

Question

Suppose a reaction has \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) values independent of temperature (i.e., assume these values are constant over some range of temperature). Show from this, and the equations given in this chapter, that $$ \ln K=\frac{-\Delta H^{\circ}}{R T}+\frac{\Delta S^{\circ}}{R} $$ where \(K\) is the equilibrium constant. How could you use values of \(K\) determined at different temperatures to determine \(\Delta H^{\circ}\) for the reaction?

Step-by-Step Solution

Answer

Use the plot of \( \ln K \) vs. \( \frac{1}{T} \) to determine \( \Delta H^{\circ} \) from the slope.

Step 1: Understanding the Gibbs Free Energy Equation

Start with the relationship for the Gibbs free energy, which is given by the equation \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \). This relates the change in free energy to enthalpy and entropy changes.

Step 2: Relating Gibbs Free Energy to the Equilibrium Constant

The connection between Gibbs free energy change and the equilibrium constant \( K \) is given by the equation \( \Delta G^{\circ} = -RT \ln K \). This expresses how free energy at standard state conditions is related to \( K \).

Step 3: Equating the Two Expressions for Gibbs Free Energy

Set \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \) equal to \( -RT \ln K \) to get the equation \( \Delta H^{\circ} - T \Delta S^{\circ} = -RT \ln K \).

Step 4: Solving for the Natural Log of the Equilibrium Constant

Rearrange the equation \( \Delta H^{\circ} - T \Delta S^{\circ} = -RT \ln K \) to solve for \( \ln K \). After rearranging, we get \( \ln K = \frac{-\Delta H^{\circ}}{RT} + \frac{\Delta S^{\circ}}{R} \).

Step 5: Using Data at Different Temperatures to Determine \( \Delta H^{\circ} \)

From the rearranged equation for \( \ln K \), plot \( \ln K \) versus \( \frac{1}{T} \). This should yield a straight line with a slope of \( \frac{-\Delta H^{\circ}}{R} \). Analyzing this slope allows us to determine \( \Delta H^{\circ} \).

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K \), quantifies the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction. It is specific to a particular reaction at a given temperature. When a reaction is at equilibrium, this constant allows us to calculate the extent of the reaction. In the context of the equation \( \Delta G^{\circ} = -RT \ln K \), the equilibrium constant becomes a vital link between Gibbs Free Energy \( \Delta G^{\circ} \) and the thermodynamic quantities of enthalpy \( \Delta H^{\circ} \) and entropy \( \Delta S^{\circ} \). By understanding the equilibrium constant, students can predict the direction of the reaction and how far it will proceed before reaching equilibrium. Moreover, since \( K \) changes with temperature, it provides a practical way to explore the temperature dependence of reaction dynamics.

Enthalpy

Enthalpy, represented by \( \Delta H \), is a measure of the total energy of a thermodynamic system. It includes internal energy plus the energy required to make room for it by displacing its environment. In chemical reactions, the change in enthalpy reflects whether a reaction is exothermic (releases heat) or endothermic (absorbs heat). In the provided exercise, enthalpy plays a crucial role in relating to Gibbs Free Energy through the equation \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \). If \( \Delta H^{\circ} \) is negative, the reaction tends to be spontaneous, assuming entropy change does not significantly counteract it. The constant value of enthalpy over a certain temperature range, as given in the problem, implies that the heat content difference between reactants and products remains unchanged, which simplifies our calculations.

Entropy

Entropy, symbolized by \( \Delta S \), is a measure of the disorder, or randomness, in a system. In thermodynamics, increase in entropy, or positive \( \Delta S \), usually favors the spontaneity of a process. In the equation \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \), entropy interacts with enthalpy and temperature to influence the free energy change. The entropy change is considered constant in this context, which simplifies finding a relationship between temperature and equilibrium constant. Ever wonder why ice melts? Well, that's because the entropy, or disorder, increases as ice becomes water, making the process favorable even if it requires input of heat.

Temperature Dependence

Temperature dependence illustrates how the behavior of a chemical reaction changes when the temperature changes. It's captured in the Gibbs Free energy equation where the term \( -T \Delta S^{\circ} \) shows a direct relation with temperature (\( T \)).When temperature changes, reaction spontaneity changes since \( \Delta G^{\circ} \) is temperature-dependent. In the equation \( \ln K = \frac{-\Delta H^{\circ}}{RT} + \frac{\Delta S^{\circ}}{R} \), temperature (\( T \)) is in the denominator of the first term indicating that as the temperature rises, the impact of the enthalpy term becomes smaller. This characteristic helps predict how likely a reaction proceeds at different temperatures. By plotting \( \ln K \) against \( \frac{1}{T} \), you find that the slope directly connects to \( \Delta H^{\circ} \), which allows for practical deductions about reaction dynamics and thermodynamic properties.