Question

The formation constant, \(K_{i}\) for the reaction $$\mathrm{Ag}^{+}(\mathrm{aq})+2 \mathrm{NH}_{3}(\mathrm{aq}) \rightleftharpoons\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}(\mathrm{aq})$$ is \(1.1 \times 10^{7} .\) What is the value of \(\Delta_{\mathrm{r}} \mathrm{G}^{\circ}\) for this reaction? Is the reaction product- or reactant- favored at equilibrium?

Short answer

\(\Delta_{r} G^{\circ} = -40.3 \text{ kJ/mol}\); the reaction is product-favored.

Step-by-step solution

Understand the relationship between K_i and Δ_r G°

The relationship between the formation constant \(K_i\) and the standard Gibbs free energy change \(\Delta_{r} G^{\circ}\) for a reaction can be given by the formula:\[ \Delta_{r} G^{\circ} = -RT \ln K_i \] where \(R\) is the universal gas constant (8.314 J/mol·K) and \(T\) is the temperature in Kelvin, typically 298 K for standard conditions.

Substitute values into the formula

We know that \(K_i = 1.1 \times 10^7\), \(R = 8.314 \text{ J/mol·K}\), and \(T = 298 \text{ K}\). Plug these values into the equation: \[ \Delta_{r} G^{\circ} = - (8.314) \times (298) \times \ln(1.1 \times 10^7) \]

Calculate natural logarithm

Calculate \(\ln(1.1 \times 10^7)\) using a calculator. The result is approximately 16.213.

Calculate Δ_r G°

Substitute the logarith mic value back into the equation: \[ \Delta_{r} G^{\circ} = - (8.314) \times (298) \times 16.213 \] Perform the multiplication: \[ \Delta_{r} G^{\circ} \approx - (8.314) \times (298) \times 16.213 \approx -40.3 \times 10^3 \text{ J/mol} \] or \(-40.3 \text{ kJ/mol}\).

Interpret the result

The calculated \(\Delta_{r} G^{\circ}\) is \(-40.3 \text{ kJ/mol}\), which is less than zero, indicating that the reaction is spontaneous under standard conditions. Therefore, the reaction is product-favored at equilibrium.

Key concepts

Gibbs Free Energy

Gibbs Free Energy, denoted as \( \Delta G \), is an essential concept in chemistry that helps us determine whether a reaction will occur spontaneously. It combines the concepts of enthalpy, entropy, and temperature into a single value. The reaction Gibbs Free Energy change, \( \Delta_{r} G^{\circ} \), tells us how a chemical reaction progresses under standard conditions. When \( \Delta_{r} G^{\circ} \) is negative, this suggests the reaction releases free energy and moves towards the formation of products, which means it is product-favored at equilibrium.
In our case, the relationship between Gibbs Free Energy and the formation constant \( K_i \) is given by the formula \( \Delta_{r} G^{\circ} = -RT \ln K_i \), where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. This equation shows a direct link: as the formation constant grows larger, \( \Delta_{r} G^{\circ} \) becomes more negative, indicating spontaneity and a preference for products.
This concept is vital in predicting and understanding how a reaction proceeds, helping chemists anticipate which direction a reaction will take given a set of conditions.

Spontaneity

Spontaneity in chemical reactions refers to the natural tendency of a reaction to proceed without external input, indicating whether a reaction is likely to occur by itself. This concept is closely tied to Gibbs Free Energy. If \( \Delta_{r} G^{\circ} \) is negative, it indicates that the reaction will be spontaneous under standard conditions.
For the reaction in our exercise, we found \( \Delta_{r} G^{\circ} \) to be \(-40.3 \text{ kJ/mol}\), which is less than zero. This means that the reaction occurs spontaneously, and does so by favoring the formation of products over reactants. Spontaneity reflects how energy transformations within the reaction align with entropy and enthalpy changes.
Key aspects to remember about spontaneity involve:

• Negative \( \Delta G \): Reaction is spontaneous and product-favored.
• Positive \( \Delta G \): Reaction is non-spontaneous and reactant-favored unless conditions change.
• Zero \( \Delta G \): Reaction is at equilibrium, with no net change in the concentrations of reactants and products.

Understanding spontaneity aids in predicting reaction behavior and planning practical applications in chemical processes.

Equilibrium Reactions

Equilibrium reactions are a fundamental concept in chemistry where reactants convert into products and vice versa at the same rate, resulting in no net change in concentration over time. The state of equilibrium indicates that the forward and reverse reactions occur continuously, maintaining a balanced dynamic.
A critical element in equilibrium reactions is the equilibrium constant \( K \), which quantifies the ratio of concentrations of products and reactants when the system is at equilibrium. In our given exercise, the formation constant, \( K_{i} \), signifies how strongly products are favored in comparison to reactants in an equilibrium state. A large \( K_{i} \) value, such as \( 1.1 \times 10^7 \), reveals a pronounced preference for the product side.
This can be summarized with the following insights:

• Large \( K \) (>>1): Product-favored equilibrium, supporting spontaneous formation of products.
• Small \( K \) (<<1): Reactant-favored equilibrium, indicating limited formation of products.
• Reversible nature: Both sides can convert into each other, enabling flexible response to changes.

Understanding equilibrium reactions enables prediction of how a system amplifies or limits product formation through changes in conditions or concentrations.