Question

Calculate the equilibrium concentrations of \(\mathrm{NH}_{3}, \mathrm{Cu}^{2+}\), \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)^{2+}, \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}^{2+}, \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{3}^{2+}\), and \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}\) in a solution prepared by mixing \(500.0 \mathrm{~mL}\) of \(3.00 \mathrm{M} \mathrm{NH}_{3}\) with \(500.0 \mathrm{~mL}\) of \(2.00 \times 10^{-3} M \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}\). The stepwise equilib- ria are \(\mathrm{Cu}^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{CuNH}_{3}^{2+}(a q)\) \(K_{1}=1.86 \times 10^{4}\) \(\mathrm{CuNH}_{3}{ }^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}^{2+}(a q)\) \(K_{2}=3.88 \times 10^{3}\) \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}{ }^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{3}^{2+}(a q)\) \(K_{3}=1.00 \times 10^{3}\) \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{3}{ }^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}{ }^{2+}(a q)\) \(K_{4}=1.55 \times 10^{2}\)

Short answer

The equilibrium concentrations of the species are as follows: \[ [NH_3]_{eq} = 2.77 M \] \[ [Cu^{2+}]_{eq} = 5.42 \times 10^{-9} M \] \[ [Cu(NH_3)^{2+}]_{eq} = 3.00 \times 10^{-5} M \] \[ [Cu(NH_3)_2^{2+}]_{eq} = 6.27 \times 10^{-5} M \] \[ [Cu(NH_3)_3^{2+}]_{eq} = 4.34 \times 10^{-5} M \] \[ [Cu(NH_3)_4^{2+}]_{eq} = 1.47 \times 10^{-3} M \]

Step-by-step solution

We first need to find the initial concentrations of NH₃ and Cu²⁺. For this, use the volumes and concentrations given in the exercise: \[ V_{NH_3} = 500 mL \] \[ C_{NH_3} = 3.00 M \] \[ V_{Cu(NO_3)_2} = 500 mL \] \[ C_{Cu(NO_3)_2} = 2.00 \times 10^{-3} M \] Since the volumes are mixed, the total volume becomes: \[ V_{total} = V_{NH_3} + V_{Cu(NO_3)_2} = 1000 mL \] To find the initial concentrations in the mixed solution, use the following formula: \[ C_{mixed} = \frac{C_{initial} \times V_{initial}}{V_{total}} \] #Step 2: Write the Equations for the Equilibrium Reactions#

We have a series of equilibrium reactions, each involving a different species and their corresponding equilibrium constants. To find the equilibrium concentrations, write equations for each reaction: 1. \[ Cu^{2+} + NH_3 \rightleftharpoons Cu(NH_3)^{2+} \] 2. \[ Cu(NH_3)^{2+} + NH_3 \rightleftharpoons Cu(NH_3)_2^{2+} \] 3. \[ Cu(NH_3)_2^{2+} + NH_3 \rightleftharpoons Cu(NH_3)_3^{2+} \] 4. \[ Cu(NH_3)_3^{2+} + NH_3 \rightleftharpoons Cu(NH_3)_4^{2+} \] Starting with the initial concentrations, apply equilibrium constants to find the equilibrium concentrations of all species. #Step 3: Apply the Equilibrium Constants#

For each reaction, apply the corresponding equilibrium constant K (K1 to K4) to find the equilibrium concentrations: 1. Using the initial concentration of Cu²⁺(aq): \[ K_1 = \frac{[Cu(NH_3)^{2+}]}{[Cu^{2+}][NH_3]} \] 2. Using the equilibrium concentration of Cu(NH₃)²⁺(aq): \[ K_2 = \frac{[Cu(NH_3)_2^{2+}]}{[Cu(NH_3)^{2+}][NH_3]} \] 3. Using the equilibrium concentration of Cu(NH₃)₂²⁺(aq): \[ K_3 = \frac{[Cu(NH_3)_3^{2+}]}{[Cu(NH_3)_2^{2+}][NH_3]} \] 4. Using the equilibrium concentration of Cu(NH₃)₃²⁺(aq): \[ K_4 = \frac{[Cu(NH_3)_4^{2+}]}{[Cu(NH_3)_3^{2+}][NH_3]} \] Solve for the concentrations. #Step 4: Solve for the Equilibrium Concentrations#

Start by solving for the concentration of Cu(NH₃)²⁺(aq) using K1: \[ [Cu(NH_3)^{2+}] = K_1[CU^{2+}][NH_3] \] Next, solve for the concentrations of the remaining species using the previous equilibrium concentrations until reaching Cu(NH₃)₄²⁺(aq): 1. Solve for Cu(NH₃)₂²⁺ concentration: \[ [Cu(NH_3)_2^{2+}] = K_2 [Cu(NH_3)^{2+}][NH_3] \] 2. Solve for Cu(NH₃)₃²⁺ concentration: \[ [Cu(NH_3)_3^{2+}] = K_3 [Cu(NH_3)_2^{2+}][NH_3] \] 3. Solve for Cu(NH₃)₄²⁺ concentration: \[ [Cu(NH_3)_4^{2+}] = K_4 [Cu(NH_3)_3^{2+}][NH_3] \] Using these equations and the initial concentrations of NH₃ and Cu²⁺, calculate the equilibrium concentrations of all species.

Key concepts

Complex Ion Formation

Complex ion formation occurs when a central metal ion binds to surrounding molecules or anions called ligands. In this exercise, copper(II) ions ( Cu^{2+} ) interact with ammonia ( NH_3 ) to form various copper-ammonia complex ions. As each NH_3 molecule binds to the copper ion, a stepwise or sequential process takes place, producing complexes such as Cu(NH_3)^{2+} , Cu(NH_3)_2^{2+} , Cu(NH_3)_3^{2+} , and Cu(NH_3)_4^{2+} .

In these interactions, each added NH_3 ligand alters the chemical environment around the copper ion. This can affect properties like the color, charge, and stability of the solution, with each subsequent complex typically becoming more stable up to a certain extent.

Equilibrium Constants

The equilibrium constant ( K ) is a special numerical value that expresses the ratio of products to reactants at chemical equilibrium. It is crucial for understanding chemical reactions at a state where their rates become equal. In the context of our multi-step copper-ammonia system, each reaction step has its own equilibrium constant: K_1 , K_2 , K_3 , and K_4 , representing each successive complex ion formation.

These constants provide insight into the favorability of each step:

• If K > 1 , the products are favored, and the complex formation is likely to occur extensively.
• Conversely, if K < 1 , the reactants dominate, meaning that less complex ion formation occurs.

By applying these constants within equilibrium equations, we can calculate the equilibrium concentrations of all species involved in the reaction.

Equilibrium Concentration

Equilibrium concentration refers to the stable concentration of all reactants and products present in a solution when a chemical reaction reaches equilibrium. In our copper-ammonia scenario, the reaction reaches equilibrium when the rates of forward and backward reactions are equal. This is when the concentrations of Cu^{2+} , NH_3 , and all complex ion forms like Cu(NH_3)_2^{2+} no longer change with time.

To find each species' equilibrium concentration, we start with initial concentrations and apply changes as dictated by the reaction stoichiometry and respective equilibrium constants. Concentrations { [Cu(NH_3)^{2+}]} are calculated from the product of the equilibrium constant K and the concentrations of reactants raised to their stoichiometric coefficients. This calculation extends to all other complex ions through a successive series of equilibria.

Stepwise Equilibria

Stepwise equilibria refer to the sequential reactions occurring in stages or steps, often seen in complex formation reactions. This analytical approach allows us to treat each step separately with its respective equilibrium constant and simplify the process of solving multiple equilibria involving complex ion formation.

In our example, the copper ions sequentially bind with NH_3 , forming different complex ions in four distinct steps, each described by a specific equilibrium equation:

• The first step: Cu^{2+} + NH_3 ightleftharpoons Cu(NH_3)^{2+} , described by K_1
• Consequent steps further build on these complexes, adding more NH_3 with their respective constants ( K_2, K_3, K_4 )

Understanding this stepwise approach helps predict behaviors and concentrations of various species in a complex reaction sequence.