Ag+(aq) + NH3(aq) ⇄ Ag(NH3)+(aq) Keq1 = 2.0 × 10³ Ag(NH3)+(aq) + NH3(aq) ⇄ Ag(NH3)2+(aq) Keq2 = 8.0 — Equilibrium Chemistry Question

STEPS:

1. Understand how to combine consecutive equilibrium reactions:
The overall chemical equation representing the formation of the diamminesilver(I) complex is:
$$\text{Ag}^+(aq) + 2\ \text{NH}_3(aq) \rightleftharpoons \text{Ag(NH}_3)_2^+(aq) \quad K_{eq3} = ?$$
This net equation is the sum of two successive, individual elementary steps:
* Step 1: $\text{Ag}^+(aq) + \text{NH}_3(aq) \rightleftharpoons \text{Ag(NH}_3)^+(aq) \quad K_{eq1} = 2.0 \times 10^3$
* Step 2: $\text{Ag(NH}_3)^+(aq) + \text{NH}_3(aq) \rightleftharpoons \text{Ag(NH}_3)_2^+(aq) \quad K_{eq2} = 8.0 \times 10^3$

2. Apply the mathematical rule for combining equilibrium constants:
When individual chemical equations are added together to produce a net reaction, their corresponding equilibrium constants must be multiplied to find the overall equilibrium constant:
$$K_{eq3} = K_{eq1} \times K_{eq2}$$
$$K_{eq3} = (2.0 \times 10^3) \times (8.0 \times 10^3) = \mathbf{1.6 \times 10^7}$$

3. Interpret the magnitude of the calculated $K_{eq3}$:
An equilibrium constant of $1.6 \times 10^7$ is extremely large ($K_{eq3} \gg 1$). This indicates that the forward reaction is highly thermodynamically favored (strongly product-favored). At equilibrium, the system will lie heavily to the right, meaning almost all available silver ions should be converted into the final complex, $\text{Ag(NH}_3)_2^+(aq)$.

4. Verify if the ligand ($\text{NH}_3$) is in excess:
To ensure the reaction can actually proceed to completion, we must check if there is enough ammonia to bind the silver:
* Equal volumes of $0.1\text{ M AgNO}_3(aq)$ and $2.0\text{ M NH}_3(aq)$ are mixed.
* Mixing equal volumes doubles the total volume of the solution, which halves the initial concentration of both species:
* $[\text{Ag}^+]_{\text{initial}} = 0.05\text{ M}$
* $[\text{NH}_3]_{\text{initial}} = 1.0\text{ M}$
* According to the $1:2$ stoichiometry of the net reaction, converting $0.05\text{ M}$ of $\text{Ag}^+$ completely requires only $0.10\text{ M}$ of $\text{NH}_3$.
* Since we have $1.0\text{ M}$ of $\text{NH}_3$ initially, ammonia is present in massive excess.

5. Conclude which species has the highest concentration:
Because there is a highly favorable overall equilibrium constant ($K_{eq3} = 1.6 \times 10^7$) and a large excess of the ligand to drive the reaction forward, nearly all silver will be converted into the stable complex. Thus, $\text{Ag(NH}_3)_2^+(aq)$ will have the highest concentration of any silver-containing species, which matches Option C.

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WHY_OTHERS_WRONG:

• Option A is incorrect: This option incorrectly claims that the dominant species is free $\text{Ag}^+(aq)$ and calculates $K_{eq3}$ as $4$. A student might get $4$ by erroneously dividing the two constants ($8.0 \times 10^3 / 2.0 \times 10^3 = 4$) rather than multiplying them. Furthermore, if $K_{eq3}$ were indeed $4$, the reaction would not go nearly as close to completion, but $\text{Ag(NH}_3)_2^+(aq)$ would still not be completely suppressed.
• Option B is incorrect: This option claims that reactant $\text{Ag}^+(aq)$ is the most abundant species at equilibrium. Because the individual step constants ($10^3$) and the combined constant ($10^7$) are so large, the concentration of uncomplexed free $\text{Ag}^+$ at equilibrium will be exceedingly small.
• Option D is incorrect: While this option correctly identifies $\text{Ag(NH}_3)_2^+(aq)$ as the dominant species, its explanation ($K_{eq1} < K_{eq2}$) is conceptually incomplete. The dominance of the final complex is dictated by the massive overall magnitude of the combined formation constant ($K_{eq3} = 1.6 \times 10^7$), not by the relative rates or strengths of the two individual steps.