Question

How is the equilibrium constant related to the standard free energy change for a reaction? (Write the equation.)

Short answer

\( \Delta G^\circ = -RT\ln(K_{eq}) \)

Step-by-step solution

Understanding the Relationship

The equilibrium constant, denoted as Keq, is directly related to the standard free energy change, denoted as ΔG°, for a reaction at a constant temperature.

Writing the Equation

The equation that relates the equilibrium constant to the standard free energy change is given by the formula: \( \Delta G^\circ = -RT\ln(K_{eq}) \) where \( \Delta G^\circ \) is the standard free energy change, R is the universal gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and \( K_{eq} \) is the equilibrium constant.

Understanding the Constants

The gas constant R and temperature T are constants under standard conditions. Thus, the equation shows that the standard free energy change is inversely proportional to the natural logarithm of the equilibrium constant; as the equilibrium constant increases, the standard free energy decreases, and vice versa.

Key concepts

Equilibrium Constant (Keq)

The equilibrium constant, represented as Keq, is a fundamental concept in the study of chemical reactions and thermodynamics. It quantifies the relative amounts of reactants and products in a reaction mixture when the reaction has reached equilibrium. Equilibrium is the state in which the rate of the forward reaction equals the rate of the reverse reaction, so there is no net change in the concentrations of reactants and products over time.

Keq is calculated using the concentrations of the products raised to the power of their stoichiometric coefficients, divided by the concentrations of the reactants raised to the power of their coefficients. For a generic reaction where A and B react to form C and D, the equilibrium constant would be expressed as:
\[ K_{eq} = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}} \]
where the square brackets denote concentration and a, b, c, and d represent the coefficients from the balanced chemical equation. Understanding Keq is vital because it informs us about the extent of the reaction, predicting if the reactants or products are favored in the equilibrium state.

Standard Free Energy Change (ΔG°)

The standard free energy change, denoted by ΔG°, is an essential thermodynamic quantity that indicates the spontaneity of a chemical reaction under standard conditions (1 bar pressure and concentrations of 1 M). A negative ΔG° means a reaction is spontaneous, meaning it can proceed without added energy. If ΔG° is positive, the reaction is non-spontaneous and requires energy input to proceed.

ΔG° is influenced by factors such as temperature, pressure, and the nature of reactants and products. It provides a window into the inherent energy changes during a reaction, combining enthalpic (ΔH°, heat change) and entropic (ΔS°, disorder change) contributions to predict the feasibility of a reaction:
\[ ΔG^\text{°} = ΔH^\text{°} - TΔS^\text{°} \]
The direct relations between ΔG°, enthalpy, and entropy play a pivotal role in understanding how energy is managed and converted during chemical reactions. This understanding is crucial for applications ranging from biological processes to industrial chemical manufacturing.

Thermodynamics in Chemistry

Thermodynamics is the branch of physical chemistry that deals with the energy and work aspects of chemical and physical processes. It provides the framework for understanding how energy is transferred and transformed. There are four laws of thermodynamics that define these principles, and they have vast implications for the study of chemistry, physics, and beyond.

In the context of chemical reactions, thermodynamics helps us predict whether processes will occur spontaneously, assessing the energy and entropy changes that accompany these reactions. The interplay of energy and matter determines the direction and extent of chemical transformations. Moreover, understanding thermodynamics is crucial when exploring the conditions under which reactions are most efficient, such as in the optimization of industrial chemical processes or the regulation of metabolic pathways in biological systems.

Gibbs Free Energy Equation

The Gibbs free energy equation bridges the realms of thermodynamics and kinetics, offering a quantifiable approach to predict the spontaneity and equilibrium position of chemical reactions. It incorporates both the enthalpy and entropy of a system, represented by the formula:
\[ G = H - TS \]
where G is the Gibbs free energy, H is the enthalpy, T is the absolute temperature in Kelvin, and S is the entropy of the system.

For reactions at equilibrium and under standard conditions, the Gibbs free energy is related to the equilibrium constant Keq by the equation previously mentioned:
\[ ΔG^\text{°} = -RT\text{ln}(K_{eq}) \]
This relationship highlights that a large Keq, which indicates a product-favored reaction, corresponds to a more negative ΔG°, implying a spontaneous reaction under standard conditions. Here, R is the universal gas constant, and ln is the natural logarithm. This powerful equation connects the macroscopic properties of thermodynamics to the microscopic view of chemical reactions, allowing chemists to calculate the essential thermodynamic parameters from experimental data.