Question

The correct relationship between free energy change in a reaction and the corresponding equilibrium constant \(K_{c}\) is (a) \(\Delta \mathrm{G}=\mathrm{RT}\) In \(\mathrm{K}\) (b) \(-\Delta \mathrm{G}=\mathrm{RT}\) In \(\mathrm{K}\) (c) \(\Delta \mathrm{G}^{\circ}=\mathrm{RT} \operatorname{In} \mathrm{K}_{\mathrm{c}}\) (d) \(-\Delta \mathrm{G}^{\circ}=\mathrm{RT} \operatorname{In} \mathrm{K}_{\mathrm{c}}\)

Short answer

The correct relationship is option (d) \( -\Delta G^\circ = RT \ln K_c \).

Step-by-step solution

Understand the Relationship

The free energy change \( \Delta G \) in a chemical reaction relates to the equilibrium constant \( K_c \) using the equation \( \Delta G = \Delta G^\circ + RT \ln Q \), where \( Q \) is the reaction quotient. At equilibrium, \( \Delta G = 0 \) and \( Q = K_c \), simplifying to \( \Delta G^\circ = -RT \ln K_c \).

Recognize Standard Conditions

The symbol \( \Delta G^\circ \) indicates the standard free energy change, which is used under standard conditions (1 atm, 298 K). The expression involving \( \Delta G^\circ \) is used to find the equilibrium constant under these conditions.

Analyze the Given Options

Review options to determine which matches the derived equation. Options (c) \( \Delta G^\circ = RT \ln K_c \) and (d) \( -\Delta G^\circ = RT \ln K_c \) correspond to relationships involving equilibrium constant. Compare them to the derived form \( \Delta G^\circ = -RT \ln K_c \).

Select the Correct Relationship

Based on the derived formula \( \Delta G^\circ = -RT \ln K_c \), option (d) \( -\Delta G^\circ = RT \ln K_c \) perfectly matches, considering the sign re-arrangement of the terms.

Key concepts

Gibbs free energy

Gibbs free energy, represented as \( \Delta G \), is a fundamental concept in chemistry that helps us predict the spontaneity of a reaction. Simply put, it tells us if a reaction will proceed on its own once it starts. A negative change in Gibbs free energy (\( \Delta G < 0 \)) indicates that the reaction is spontaneous, or favorable, under the given conditions.
On the other hand, a positive \( \Delta G \) means the reaction is non-spontaneous, and will need energy input to proceed.

A key equation related to Gibbs free energy is:
\[ \Delta G = \Delta G^\circ + RT \ln Q \]
Here:

• \( \Delta G^\circ \): Standard Gibbs free energy change
• \( R \): Universal gas constant
• \( T \): Temperature in Kelvin
• \( Q \): Reaction quotient

At equilibrium, the reaction quotient \( Q \) equals the equilibrium constant \( K_c \), and \( \Delta G \) becomes zero.
This simplifies to \( \Delta G^\circ = -RT \ln K_c \), showing a direct relationship between Gibbs free energy change under standard conditions and the equilibrium constant.

equilibrium constant

The equilibrium constant, \( K_c \), is a critical number in chemistry that indicates the ratio of concentrations of products to reactants at equilibrium.
It is a measure of the extent to which a reaction proceeds before reaching equilibrium. A large \( K_c \) indicates that at equilibrium, products are favored, and the reaction essentially proceeds to completion.
Conversely, a small \( K_c \) suggests that reactants are favored.

The nature of \( K_c \) and its impact allows chemists to predict the direction of a chemical reaction and understand its possible outcomes.
It connects closely with Gibbs free energy, helping us determine the favorability and feasibility of reactions under specific conditions.

The relationship \( \Delta G^\circ = -RT \ln K_c \) highlights this connection, as \( \Delta G^\circ \) represents the free energy change at standard conditions.
This equation effectively links the thermodynamic properties of a reaction with its equilibrium behavior. When applied correctly, it offers valuable insights into how a reaction behaves given certain conditions.

standard conditions in thermodynamics

Standard conditions in thermodynamics are a set of specific conditions under which measurements are made to ensure consistency.
Under these conditions, the pressure is set at 1 atmosphere (atm), and the temperature is 298 K (25 °C).
These baseline conditions are important because they provide a common reference point to compare different chemical reactions.

When we talk about \( \Delta G^\circ \), we refer to the Gibbs free energy change under these standard conditions.
By having a defined baseline, scientists can predict the outcome of reactions accurately and consistently across various studies.
This makes \( \Delta G^\circ \) an invaluable tool in chemical thermodynamics as it allows comparisons and calculations across different experiments and data sets.

Using the equation \( \Delta G^\circ = -RT \ln K_c \), calculations under standard conditions become straightforward, allowing for easier prediction and understanding of chemical behavior.
It is this standardization that helps facilitate clear communication and understanding among chemists on a global scale.