In this section, we will delve into the intricate relationship between entropy, equilibrium, and Gibbs energy, three fundamental concepts in thermodynamics. Understanding these concepts is crucial for predicting how chemical reactions proceed and determining the conditions under which they reach equilibrium.

**ΔG and Reaction Equilibrium**

Gibbs free energy (G) is a thermodynamic potential that combines enthalpy, entropy, and temperature to predict the spontaneity of a reaction. The change in Gibbs free energy (ΔG) during a reaction provides invaluable information about the reaction's behaviour concerning equilibrium.

**Spontaneous Reactions:**If ΔG is negative, the reaction is spontaneous and will proceed in the forward direction until it reaches equilibrium.**Non-Spontaneous Reactions:**If ΔG is positive, the reaction is non-spontaneous, and the reverse reaction is favoured.**Equilibrium:**At equilibrium, ΔG is zero, meaning there is no net change in the system's Gibbs free energy.

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**Calculating ΔG**

The change in Gibbs free energy during a reaction can be calculated using the equation:

ΔG = ΔG⦵ + RT lnQ

where:

- ΔG⦵ is the standard Gibbs free energy change,
- R is the gas constant (8.314 J/(mol·K)),
- T is the temperature in Kelvin,
- Q is the reaction quotient, which is a ratio of product activities to reactant activities.

**Relationship with Equilibrium Constant**

At equilibrium, Q becomes the equilibrium constant (K), and ΔG is zero. Substituting these values into the equation gives:

0 = ΔG⦵ + RT lnK

Rearranging the equation provides a way to calculate the standard Gibbs free energy change from the equilibrium constant:

ΔG⦵ = −RT lnK

This equation highlights the direct relationship between ΔG⦵ and K, demonstrating that knowing the value of one allows us to calculate the other.

**Predicting Equilibrium Composition**

The standard Gibbs free energy change (ΔG⦵) also provides insights into the composition of the equilibrium mixture.

**Negative ΔG⦵:**If ΔG⦵ is negative, the equilibrium constant (K) is greater than one, indicating that products are favoured at equilibrium.**Positive ΔG⦵:**If ΔG⦵ is positive, K is less than one, suggesting that reactants are favoured.**ΔG⦵ Equals Zero:**This indicates that the system is perfectly at equilibrium, with no net change in the concentrations of reactants and products.

Image courtesy of The Chemistry Notes

**Application in Calculations**

When predicting the composition of an equilibrium mixture, it is crucial to understand how changes in conditions such as temperature and pressure affect ΔG⦵ and, consequently, the position of equilibrium.

**Effect of Temperature:**An increase in temperature will increase the value of RT lnK, potentially changing the sign of ΔG⦵ and shifting the equilibrium position.**Effect of Pressure:**For reactions involving gases, changes in pressure can affect the equilibrium position by altering the concentration of reactants and products, thus changing Q and potentially ΔG⦵.

**Example Calculations**

**Calculating K from ΔG⦵**

Given a standard Gibbs free energy change (ΔG⦵) of -30 kJ/mol at 298 K, calculate the equilibrium constant (K).

- Convert ΔG⦵ to J/mol: -30 kJ/mol × 1000 J/kJ = -30,000 J/mol
- Substitute values into the equation ΔG⦵ = −RT lnK: -30,000 J/mol = −(8.314 J/(mol·K))(298 K) lnK
- Solve for lnK: lnK = 11.33
- Find K: K = e
^{11.33}≈ 83000

**Predicting Equilibrium Position**

Consider a reaction with a ΔG⦵ of 10 kJ/mol at 350 K. What can we infer about the equilibrium position?

- The positive ΔG⦵ indicates that K is less than one, meaning that reactants are favoured at equilibrium. This reaction is non-spontaneous under the given conditions.

In summary, understanding the relationship between ΔG, ΔG⦵, and equilibrium is pivotal in predicting how a reaction will proceed under different conditions and determining the composition of the equilibrium mixture. By mastering these concepts and equations, students can confidently analyse and predict the behaviour of chemical reactions.

## FAQ

Yes, the value of ΔG can change as a reaction progresses. While ΔG⦵ is constant for a given reaction at a specific temperature, ΔG is affected by the concentrations of reactants and products, represented by the reaction quotient, Q. As these concentrations change over the course of the reaction, so does Q, leading to a changing ΔG value. This is reflected in the equation ΔG = ΔG⦵ + RT lnQ.

The sign of ΔG⦵ provides insight into which direction a reaction will spontaneously proceed. A negative ΔG⦵ indicates a spontaneous forward reaction, and in this case, K > 1, meaning products are favoured at equilibrium. Conversely, a positive ΔG⦵ suggests the reverse reaction is spontaneous, with K < 1, indicating reactants are favoured at equilibrium. A ΔG⦵ of zero, as previously mentioned, means the system is already at equilibrium and K = 1.

The equation ΔG = ΔG⦵ + RT lnQ is used to determine the Gibbs free energy change for a system that is not necessarily at equilibrium. Q, the reaction quotient, represents the current state of the system, whereas K, the equilibrium constant, represents the state when the system is at equilibrium. When Q = K, the system is at equilibrium and ΔG = 0. The equation thus allows for the assessment of the direction and extent to which a reaction will proceed to reach equilibrium from any given starting point.

Temperature plays a crucial role in determining the spontaneity of a reaction. The equation ΔG⦵ = ΔH⦵ − TΔS⦵ highlights this relationship. If ΔH⦵ is negative (exothermic) and ΔS⦵ is positive, then ΔG⦵ will always be negative, making the reaction spontaneous at all temperatures. However, if ΔH⦵ and ΔS⦵ have opposite signs, the spontaneity is determined by the magnitude of TΔS⦵ relative to ΔH⦵. In some cases, reactions that are non-spontaneous at low temperatures may become spontaneous at higher temperatures, and vice versa.

When ΔG⦵ is zero, it means that the system is at equilibrium at the given conditions. Neither the forward nor the reverse reaction is favoured, and the concentration of reactants and products remains constant over time. This state represents the maximum entropy or randomness that can be achieved without any net change in the system. The value of the equilibrium constant, K, under such conditions would be 1, indicating equal concentrations of products and reactants.

## Practice Questions

To find the equilibrium constant (K), we can use the equation ΔG⦵ = −RT lnK. First, convert ΔG⦵ to J/mol: +20 kJ/mol × 1000 J/kJ = +20,000 J/mol. Next, substitute the values into the equation:

+20,000 J/mol = −(8.314 J/(mol·K))(298 K) lnK

Solve for lnK:lnK = -8.01

Find K:

K = e^{(-8.01)} ≈ 3.35 × 10^{(-4)}

So, the equilibrium constant, K, for the reaction at 298 K is approximately 3.35 × 10^{(-4)}.

To find ΔG⦵, we can rearrange the equation ΔG⦵ = −RT lnK to solve for ΔG⦵:ΔG⦵ = −(8.314 J/(mol·K))T ln(5 × 10^{(-3)})

Without the specific temperature, we can't calculate a numerical value. However, it's clear that since ln(5 × 10^{(-3)}) is negative, ΔG⦵ will be positive. A positive ΔG⦵ indicates that the reaction is non-spontaneous under the given conditions and that the reactants are favoured at equilibrium. This understanding is crucial for predicting the behaviour of chemical reactions.