The formation constant for the reaction \(\mathrm{Ag}^{+}+\) \(2 \mathrm{NH}_{3} \rightleftharpoons\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}\) is \(1.5 \times 10^{7}\) and that for the reaction \(\mathrm{Ag}^{+}+2 \mathrm{CN}^{-} \rightleftharpoons\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}\) is 1.0 \(\times 10^{21}\) at \(25^{\circ} \mathrm{C}\) (see Table 17.4 ). Calculate the equilibrium constant at \(25^{\circ} \mathrm{C}\) for the reaction $$\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}+2 \mathrm{CN}^{-} \rightleftharpoons\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}+2 \mathrm{NH}_{3}$$

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. That means, there is no net change in the concentration of reactants and products over time, even though the reactions are still occurring on a molecular level.

Understanding chemical equilibrium is crucial for not just academic purposes but also for various industries like pharmaceuticals and materials science where reactions are carefully timed and balanced. For instance, when you have a reaction like \(Ag^{+} + 2NH_{3} \rightleftharpoons [Ag(NH_{3})_{2}]^{+}\), equilibrium is reached when the rate at which the silver ion and ammonia combine to form its complex equals the rate at which the complex decomposes back into silver ions and ammonia.

To make sense of this in a real-world context, consider a crowded room where people are constantly moving in and out. If the number of people entering matches the number leaving, the total number inside the room stays the same — this is analogous to a chemical system at equilibrium.

The formation constant, also known as the stability constant, is a type of equilibrium constant that quantifies the stability of a complex in solution. Higher values of the formation constant indicate a more stable complex. The given exercise involves the formation constants of silver complexes with ammonia and cyanide, with the respective values given as \(1.5 \times 10^{7}\) and \(1.0 \times 10^{21}\).

Let's take a closer look at what these numbers mean. A formation constant of \(1.5 \times 10^{7}\) for the silver-ammonia complex tells us that once silver ions and ammonia combine to form the complex, it's quite stable — it prefers staying in that combined form instead of reverting back. Conversely, the significantly larger constant for the silver-cyanide complex indicates an even greater stability, making it much less likely to dissociate back into silver ions and cyanide.

The reaction quotient, \(Q\), is a measure at a particular moment that tells us which direction a reaction will proceed to reach equilibrium. You calculate \(Q\) using the same expression as for the equilibrium constant \(K\), but with the current concentrations (or pressures) of the reactants and products, instead of at equilibrium.

If \(Q\) is less than \(K\), the forward reaction is favored, and the system will proceed to the right to reach equilibrium. Conversely, if \(Q\) is greater than \(K\), the reverse reaction is favored, and the system will shift to the left. When \(Q\) equals \(K\), the system is at equilibrium, and no further change in concentrations will occur. In the context of our exercise, knowing the reaction quotient can help predict whether [\text{Ag}(NH_{3})_{2}]^{+} will react with \(CN^{-}\) or if [\text{Ag}(CN)_{2}]^{-} will dissociate to form \(Ag^{+}\) and \(CN^{-}\), facilitating our understanding of the direction of the equilibrium shift.