Problem 54 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Is the standard free-energy change, \(\Delta G^{\circ}\), always larger than \(\Delta G ?\) (b) For any process that occurs at constant temperature and pressure, what is the significance of \(\Delta G=0 ?\) (c) For a certain process, \(\Delta G\) is large and negative. Does this mean that the process necessarily has a low activation barrier?

Step-by-Step Solution

Answer

(a) No, ΔG° is not always larger than ΔG. Their relationship is given by the Gibbs-Helmholtz equation: \( \Delta G = \Delta G^{\circ} + RT \ln{Q} \), and ΔG can be larger or smaller than ΔG°, depending on the reaction's progress, temperature, and Q's value. (b) When ΔG = 0 for a process occurring at constant temperature and pressure, it implies that the process is in equilibrium, and neither the forward nor the reverse process is favored. (c) A large negative ΔG means that the reaction is thermodynamically favorable, but it does not necessarily imply that the reaction has a low activation barrier. These are related to reaction kinetics, represented by the activation energy (Ea) in the Arrhenius equation.

Step 1: (a) Comparison of ΔG° and ΔG

It is not true that the standard free-energy change (ΔG°) is always larger than the Gibbs free-energy change (ΔG). The relationship between ΔG and ΔG° is given by the Gibbs-Helmholtz equation: \[ \Delta G = \Delta G^{\circ} + RT \ln{Q} \] Where R is the gas constant, T is the temperature in Kelvin, and Q is the reaction quotient. As the reaction progresses, Q deviates from 1 and the value of ΔG changes. Depending on the reaction's progress, temperature, and Q's value, ΔG might be larger or smaller than ΔG°.

Step 2: (b) Significance of ΔG = 0

When ΔG = 0 for a process occurring at constant temperature and pressure, it implies that the process is in equilibrium. Neither the forward nor the reverse process is favored, and the concentrations of reactants and products remain constant. At equilibrium, the reaction quotient (Q) is equal to the equilibrium constant (K): \[ \Delta G = 0 = \Delta G^{\circ} + RT \ln{K} \] This signifies that in a reversible process, the forward and reverse reactions occur at the same rate, maintaining a constant equilibrium between reactants and products.

Step 3: (c) Relationship between ΔG and Activation Barrier

A large negative ΔG means that the reaction is thermodynamically favorable and will proceed spontaneously in the forward direction. However, it does not necessarily imply that the reaction has a low activation barrier. The activation barrier is related to the reaction kinetics, which determines the reaction rate. This activation barrier is represented by the activation energy (Ea) in the Arrhenius equation: \[ k = Ae^{\frac{-Ea}{RT}} \] Even if a reaction is thermodynamically favorable (large negative ΔG), it might have a high activation energy that requires external input (e.g., a catalyst) to lower the barrier and increase the reaction rate. So, a large negative ΔG does not provide information about the height of the activation barrier.

Key Concepts

Standard Free Energy Change

The standard free energy change, often symbolized as \( \Delta G^{\circ} \), is a crucial concept in thermodynamics. It represents the change in Gibbs free energy when a reaction goes from pure reactants to pure products under standard conditions. These conditions are typically defined as 1 atmosphere of pressure, a temperature of 25°C (298 K), and a concentration of 1 M for all solutions involved.

Standard free energy change is central to predicting the spontaneity of a chemical reaction. If \( \Delta G^{\circ} \) is negative, the reaction is likely to be spontaneous under standard conditions. On the flip side, a positive \( \Delta G^{\circ} \) suggests that the reaction is non-spontaneous as written. However, it is important to note:

• \( \Delta G^{\circ} \) is not necessarily the same as the actual free energy change \( \Delta G \) during a reaction.
• As the reaction progresses and conditions deviate from standard, the reaction quotient \( Q \) affects \( \Delta G \) in such a way that it might differ from \( \Delta G^{\circ} \).

The relationship between the two is established by the Gibbs-Helmholtz equation: \( \Delta G = \Delta G^{\circ} + RT \ln{Q} \). Understanding this equation helps explain why \( \Delta G \) can be larger or smaller than \( \Delta G^{\circ} \) depending on the particulars of the chemical process.

Equilibrium Thermodynamics

Equilibrium thermodynamics involves the study of systems that are in thermal equilibrium. In the context of chemical reactions, this is especially significant as it provides deeper insight into the balance between reactants and products when neither reaction direction is favored. This balanced state is achieved when the reaction's Gibbs free energy change, \( \Delta G \), equals zero.

When \( \Delta G = 0 \), it signifies that the system is at equilibrium. At this point, the concentrations of reactants and products remain constant, and the forward and backward reactions occur at the same rate. This equilibrium condition can be mathematically expressed as:

• \( Q = K \) where \( Q \) is the reaction quotient, and \( K \) is the equilibrium constant.
• The equation \( \Delta G = 0 = \Delta G^{\circ} + RT \ln{K} \) describes this equilibrium state.

Understanding equilibrium is vital for interpreting how external conditions like pressure and temperature could shift the balance of a reaction. It identifies scenarios under which adjustments to these conditions might push a reaction either forward to completion or backward towards reactants.

Activation Energy

Activation energy is a critical concept in understanding how reactions occur. It's the minimum energy required for a reaction to proceed. Even if a reaction is thermodynamically favorable, indicated by a large negative \( \Delta G \), it might still not happen quickly without sufficient activation energy.

The activation energy \( E_a \) influences the reaction rate, independent of the free energy change. It's a key parameter in the Arrhenius equation, which explicitly relates \( E_a \) to the rate constant \( k \):

• \( k = Ae^{-\frac{Ea}{RT}} \)
• Here, \( A \) is the pre-exponential factor, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

This equation shows how lower activation energies potentially lead to faster reactions, while higher ones can slow down the rate significantly. Therefore, catalysts play an essential role by lowering \( E_a \), allowing even those reactions with very high activation energies to proceed at a significant rate.

This concept highlights that, in reaction kinetics, a spontaneous reaction does not necessarily imply it will proceed without an external input or catalyst to overcome this barrier.