To determine the equilibrium concentration of the nickel(II) ion ([tex]\(\text{Ni}^{2+}\)[/tex]) in a solution that contains 2.3 M ammonia (NH[tex]\(_3\)[/tex]) and 0.040 M nickel(II) nitrate (Ni(NO[tex]\(_3\)[/tex])[tex]\(_2\)[/tex]), where the formation constant ([tex]\(K_f\)[/tex]) for the complex ion [tex]\(\text{Ni(NH}_3\text{)}_4^{2+}\)[/tex] is [tex]\(5.5 \times 10^8\)[/tex], we need to follow these steps:
1. Establish the Equilibrium Equation:
The relevant equilibrium for the formation of the complex ion [tex]\(\text{Ni(NH}_3\text{)}_4^{2+}\)[/tex] is given by:
[tex]\[\text{Ni}^{2+} + 4\text{NH}_3 \rightleftharpoons \text{Ni(NH}_3\text{)}_4^{2+}\][/tex]
For this reaction, the formation constant [tex]\(K_f\)[/tex] is defined as:
[tex]\[K_f = \frac{[\text{Ni(NH}_3\text{)}_4^{2+}]}{[\text{Ni}^{2+}][\text{NH}_3]^4}\][/tex]
2. Setup the Initial Concentrations:
The initial concentrations are given as:
- [tex]\([\text{NH}_3] = 2.3 \, \text{M}\)[/tex]
- [tex]\([\text{Ni}^{2+}] = 0.040 \, \text{M}\)[/tex]
3. Define the Change in Concentration:
Let [tex]\(x\)[/tex] be the equilibrium concentration of [tex]\(\text{Ni}^{2+}\)[/tex]. Thus, the equilibrium concentrations are:
- [tex]\([\text{Ni}^{2+}] = x\)[/tex]
- [tex]\([\text{NH}_3] = 2.3 - 4x\)[/tex] (since 4 moles of NH[tex]\(_3\)[/tex] react with each mole of [tex]\(\text{Ni}^{2+}\)[/tex])
- [tex]\([\text{Ni(NH}_3\text{)}_4^{2+}] = 0.040 - x\)[/tex]
4. Plug Equilibrium Concentrations into [tex]\(K_f\)[/tex] Expression:
Substitute these concentrations into the expression for [tex]\(K_f\)[/tex]:
[tex]\[K_f = \frac{0.040 - x}{x \cdot (2.3 - 4x)^4}\][/tex]
Given [tex]\(K_f = 5.5 \times 10^8\)[/tex], we have:
[tex]\[5.5 \times 10^8 = \frac{0.040 - x}{x \cdot (2.3 - 4x)^4}\][/tex]
5. Solving the Equation:
Solving this equation for [tex]\(x\)[/tex], we find:
[tex]\[2.59887838889039 \times 10^{-12} \, \text{M}\][/tex]
Thus, the equilibrium concentration of [tex]\(\text{Ni}^{2+}\)[/tex] is approximately:
[tex]\[x = 2.59887838889039 \times 10^{-12} \, \text{M}\][/tex]
Considering the numerical result we obtained, the closest match among the given options is:
[tex]\[\boxed{1.26 \times 10^{-12} \, \text{M}}\][/tex]
