From the value of \(K_{f}\) listed in Table \(17.1,\) calculate the concentration of \(\mathrm{Ni}^{2+}\) in \(1.0 \mathrm{~L}\) of a solution that contains a total of \(1 \times 10^{-3} \mathrm{~mol}\) of nickel(II) ion and that is \(0.20 \mathrm{M}\) in \(\mathrm{NH}_{3}\)

Understanding chemical equilibrium is essential when dealing with reactions that can occur in both forward and reverse directions. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, leading to constant concentrations of the reactants and products, despite both reactions still occurring. This dynamic state can be disrupted by changing conditions such as concentration, temperature, or pressure, following Le Chatelier's principle.

For example, in the context of complex ion formation, adding more ligands or metal ions can shift the equilibrium towards the product side, resulting in more complex ion formation. Conversely, removing a reactant or product will shift the equilibrium to oppose the change. It's crucial to note that equilibrium does not imply equal concentrations of reactants and products but rather a constant ratio as defined by the equilibrium constant.

The formation constant, or stability constant, denoted as K_f, quantifies the stability of a complex ion in a solution. It's a special type of equilibrium constant referring specifically to the formation of complex ions. A high K_f value indicates a stable complex ion that forms readily from its constituent ions.

For students, understanding how to use K_f is paramount in predicting whether a complex will form under certain conditions. In the case of the Ni²⁺ and NH₃ reaction, the high value of K_f allows assumptions that simplify the mathematical approach to finding equilibrium concentrations—assuming that nearly all metal ions react to form the complex, thus simplifying the algebra involved in solving for 'x'.

An equilibrium expression mathematically represents the ratio of concentration terms for products raised to the power of their stoichiometric coefficients to the reactants raised to the power of their coefficients at equilibrium. The expression for a reaction such as the formation of a complex ion is given by
\[K = \frac{[\text{Products}]}{[\text{Reactants}]^{\text{stoichiometric coefficients}}}\]
For students tackling equilibrium problems, it is crucial to write the correct equilibrium expression and understand that only species in aqueous solution or in the gaseous state are included; solids and pure liquids do not appear in this expression.

Using the correct stoichiometry as seen in the given Ni²⁺ and NH₃ reaction is vital for the expression, highlighting that the coefficients become the exponents in the expression. It underscores the balance of chemical equations before using them to write the equilibrium expression.