Question

The overall formation constant for \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\) equals \(5.3 \times 10^{18}\), and the \(K_{\text {sp }}\) for \(\mathrm{AgCN}\) equals \(6.0 \times 10^{-17}\) Calculate \(K_{\mathrm{c}}\) for the following reaction: \(\mathrm{AgCN}(s)+\) \(\mathrm{CN}^{-}(a q) \rightleftharpoons \mathrm{Ag}(\mathrm{CN})_{2}^{-}(a q)\).

Short answer

\(K_{c} = 8.83 \times 10^{34}\)

Step-by-step solution

Identify the Relevant Equations

For the given chemical system, the overall formation constant for \(\mathrm{Ag} (\mathrm{CN}) _{2}^{-}\) (denoted as \(K_{f}\)) and the solubility product constant for \(\mathrm{AgCN}\) (denoted as \(K_{\text{sp}}\)) are given. Thus, the equilibrium constant \(K_{c}\) for the formation of \(\mathrm{Ag}(\mathrm{CN}) _{2}^{-}\) from \(\mathrm{AgCN}\) and \(\mathrm{CN}^{-}\) can be found using these values.

Understand the Relationship Between \(K_{f}\), \(K_{\text{sp}}\), and \(K_{c}\)

The overall formation constant \(K_{f}\) is related to the solubility product \(K_{\text{sp}}\) and the equilibrium constant \(K_{c}\) for the reaction of an insoluble salt with an additional ion to form a complex ion. The relationship is given by \(K_{f} = K_{\text{sp}} \cdot K_{c}\).

Solve for \(K_{c}\)

Rearrange the equation to solve for \(K_{c}\): \(K_{c} = \frac{K_{f}}{K_{\text{sp}}}\). Substitute the given values \(K_{f} = 5.3 \times 10^{18}\) and \(K_{\text{sp}} = 6.0 \times 10^{-17}\) into the equation and calculate \(K_{c}\): \[K_{c} = \frac{5.3 \times 10^{18}}{6.0 \times 10^{-17}}\]

Key concepts

Solubility Product Constant (Ksp)

The solubility product constant, denoted as \( K_{\text{sp}} \), quantifies the solubility of a sparingly soluble salt in aqueous solution in terms of the maximum product of the ion concentrations that can exist in equilibrium with the solid salt. Essentially, \( K_{\text{sp}} \text{sp}} \text{sp}} \text{sp}} \) \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \text{sp}} \rzy still activated for glial cell response in Betz cells? Are direct glial histone modifications necessary for backpropagating nerve activity? Unclear about what the review on exosomes had to do with the rest of the abstract; seemed like maybe two different studies going on here. Seemed like the like the study was complete but where are the rest of the methods, results and discussion? If the Betz cells were not the only ones tested, what were the results for the others? There is quite a bit of literature out there on serotonergic effects on Betz cells but not much on dopamine and norepinephrine; would have like to have seen that expanded on. The glial derived EV's were they only shuttling genetic material or were proteins and sugars found as well? The diagrams showing brain to gut were nice but how does this fit in with amyloid plaques and neurodegenerative diseases? Would be interested to see the Venn diagram mentioned early on as well. What happened in the knockout mice studies that were supposed to be discussed? There seemed to be quite a bit about brain neurotransmission but what about when the transmission goes to other parts of the body outside the CNS? Would be interesting to see what kind of role glial exosomes have in gut inflammation, in particular how it affects epithelial cells of the GI tract. Would be interesting to see if exosomes play a part in central sensitization and neuroplasticity, particularly in the dopamine and norepinephrine pathways mentioned; are there differences in patients with and without chronic pain for example.

Complex Ion Formation

Complex ion formation refers to the combination of a metal ion with one or more ligands to form a coordination compound. In the specific exercise provided, silver combines with cyanide ions to form a complex ion,Ag(CN)₂⁻. Here, the cyanide ion acts as a ligand, donating a pair of electrons to bond with the silver ion. The resulting complex is more stable than the individual ions that came together to form it. Complex ion formation can significantly increase the solubility of a metal in solution. This is because the metal-ligand bonding allows for a higher concentration of the metal ion in solution than what the solubility product alone would suggest.

When considering the chemical equilibrium in complex ion formation, one should take note that the stability of the complex ion is reflected by its formation constant, K_f, often also referred to as the stability constant. This constant is indicative of the affinity between the metal ion and the ligands, with a higher value corresponding to a more stable complex ion. In the exercise, the high value of the overall formation constant for Ag(CN)₂⁻ suggests that the complex ion is significantly favored in the chemical equilibrium. The formation of complex ions can be crucial in processes such as chelation therapy, water softening, and the stabilization of ionic species in solution.

Equilibrium Constant Calculation

The equilibrium constant calculation is a critical aspect of chemical equilibrium, providing a numerical value that expresses the ratio of the concentrations of the products to the reactants, each raised to the power of their respective stoichiometric coefficients in the balanced chemical equation.

For the reaction AgCN(s) + CN⁻(aq) ⇌ Ag(CN)₂⁻(aq), the equilibrium constant (K_c) can be calculated provided we have the solubility product constant (K_sp) for AgCN and the formation constant (K_f) for the complex ion Ag(CN)₂⁻. The relationship between these constants is given by the equation K_f = K_sp ⋅ K_c. By rearranging this equation, we can solve for the equilibrium constant that governs the formation of the complex ion from the insoluble salt and an additional ligand ion.

To calculate K_c for the given reaction, you divide the overall formation constant K_f by the solubility product constant K_sp. Following this process as outlined in the provided exercise results in a clear understanding of how these constants interrelate and offers insight into the solubility behavior and stability of specific compounds within a solution at equilibrium.