Question

What is the relationship between the standard free-energy change, \(\Delta G^{\circ}\), for a reaction and the equilibrium constant, \(K ?\) What is the sign of \(\Delta G^{\circ}\) when: (a) \(K>1\) (b) \(K=1\) (c) \(K<1\)

Short answer

(a) \( \Delta G^{\circ} < 0 \); (b) \( \Delta G^{\circ} = 0 \); (c) \( \Delta G^{\circ} > 0 \).

Step-by-step solution

Understanding the Relationship

The relationship between the standard free-energy change, \( \Delta G^{\circ} \), and the equilibrium constant, \( K \), for a reaction at a given temperature is expressed by the equation: \[ \Delta G^{\circ} = -RT \ln K \] where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. This equation indicates that the sign and magnitude of \( \Delta G^{\circ} \) are related to the value of \( K \).

Analyzing the Case: K > 1

When \( K > 1 \), the natural logarithm, \( \ln K \), is positive, causing \( -RT \ln K \) to be negative. Hence, \( \Delta G^{\circ} < 0 \). This means the reaction is spontaneous in the forward direction under standard conditions.

Analyzing the Case: K = 1

When \( K = 1 \), the natural logarithm, \( \ln K \), equals \( 0 \). Therefore, \( \Delta G^{\circ} = 0 \). This means the reaction is at equilibrium under standard conditions, and there is no net change in free energy.

Analyzing the Case: K < 1

When \( K < 1 \), the natural logarithm, \( \ln K \), is negative, causing \( -RT \ln K \) to be positive. Thus, \( \Delta G^{\circ} > 0 \). This means the reaction is non-spontaneous in the forward direction under standard conditions.

Key concepts

Free Energy

The concept of free energy, particularly Gibbs free energy \( \Delta G^{\circ} \), is a cornerstone in thermodynamics. It helps us determine whether a chemical reaction can proceed on its own—known as spontaneity.Free energy is essentially the energy that can be converted into work at constant temperature and pressure, which is critical in chemical and biological processes.

The equation linking free energy to the equilibrium constant is: \[ \Delta G^{\circ} = -RT \ln K \]where:

• \(R\) is the universal gas constant
• \(T\) is the temperature in Kelvin
• \(K\) is the equilibrium constant

This equation shows the intimate connection between the free energy change and how a reaction is skewed towards products or reactants.By observing the sign of \( \Delta G^{\circ} \), we can quickly gauge the direction of a reaction:

• Negative \( \Delta G^{\circ} \): Reaction is spontaneous
• Zero \( \Delta G^{\circ} \): Reaction is at equilibrium
• Positive \( \Delta G^{\circ} \): Reaction is non-spontaneous

Equilibrium Constant

The equilibrium constant, \( K \), represents the ratio of concentrations of products to reactants at equilibrium for a given reaction.

It's a measure of the extent to which a reaction proceeds and helps us understand the balance between forward and reverse reactions.
Though its value doesn't concern time, it tells us about the position of equilibrium:

• \( K > 1 \): Products are favored over reactants
• \( K = 1 \): No preference; equivalent amounts of products and reactants
• \( K < 1 \): Reactants are favored over products

The relation between \( \Delta G^{\circ} \) and \( K \) using the equation \( \Delta G^{\circ} = -RT \ln K \) can further illuminate these states. For example, a large \( K\) results in a negative \( \Delta G^{\circ} \), showing spontaneous reactions towards products, while a small \( K\) suggests positive \( \Delta G^{\circ} \), indicating non-spontaneous reactions.

Spontaneity

Spontaneity refers to the ability of a chemical reaction to occur without any external energy input. It doesn't necessarily mean how fast the reaction will take place but whether it can happen on its own.

The sign of \( \Delta G^{\circ} \, \) gives us a straightforward understanding of spontaneity:For a spontaneous reaction:

• \( \Delta G^{\circ} < 0 \): Reaction will proceed on its own under standard conditions.
  This typically aligns with \( K > 1 \), suggesting that products are favored.
• \( K = 1 \): Reactants and products are balanced, and \( \Delta G^{\circ} \) equals zero,
  indicating no net energy change—equilibrium.
• \( K < 1 \): \( \Delta G^{\circ} > 0 \) shows that the reaction is non-spontaneous,
  as a result, external energy would be required for the reaction to proceed.

Understanding the connection between free energy, equilibrium constants, and spontaneity allows us to predict and control chemical processes efficiently.