Question

From the value of \(K_{f}\) listed in Table \(17.1\), calculate the concentration of \(\mathrm{Cu}^{2+}\) in \(1.0 \mathrm{~L}\) of a solution that contains a total of \(1 \times 10^{-3} \mathrm{~mol}\) of copper(II) ion and that is \(0.10 \mathrm{M}\) in \(\mathrm{NH}_{3}\).

Short answer

The concentration of Cu^2+ ions in the solution can be found by using the formation constant, Kf, and setting up an equilibrium expression. Using an ICE table, the concentration of Cu^2+ can be represented by x. The equation to solve for x is \(K_{value} = \frac{x}{(1-x)(0.10-4x)^4}\), where Kvalue is given in Table 17.1. By assuming x is much smaller than the initial concentration of NH3, we can simplify the equation and calculate x using \(x = K_{value} \times 0.10^4\). The value of x represents the concentration of Cu^2+ ions in the solution.

Step-by-step solution

Identify the given parameters

We are given the following information: - Total copper(II) ion in the solution = 1x10-3 mol - Concentration of NH3 = 0.10 M

Write the reaction for complex formation

The complex formation reaction between copper(II) ions (Cu^2+) and ammonia (NH3) is: \[Cu^{2+}(aq) + 4NH_3(aq) \rightleftharpoons Cu(NH_3)_4^{2+}(aq)\] The formation constant for this reaction is Kf, which is given in Table 17.1.

Set up the equilibrium expressions and ICE table

Now, we can set up the equilibrium expressions for this reaction. By applying the law of mass action, we have: \[K_f = \frac{[Cu(NH_3)_4^{2+}]}{[Cu^{2+}][NH_3]^4}\] Next, we set up an ICE table (Initial, Change, Equilibrium) to calculate the concentrations of the species at equilibrium. | | Cu^2+ | NH3 | Cu(NH_3)_4]^2+ | |------|-------|-------|---------------| | I | x | 0.10 | 0 | | C | -x | -4x | +x | | E | 0 | 0.10-4x | x | Here, x represents the concentration of Cu^2+ ions in the solution and the changes in concentration of the other species during the reaction.

Plug in the values and solve for x

Now we can plug the equilibrium concentrations from the ICE table into the Kf expression: \[K_f = \frac{x}{(1-x)(0.10-4x)^4}\] Substitute the given value of Kf (available in Table 17.1) and solve for x: Since the exact value of Kf is not provided in the problem, I'll denote it as a placeholder (Kvalue). \(K_{value} = \frac{x}{(1-x)(0.10-4x)^4}\) To solve for x, we need to rearrange the equation and then use numerical methods or approximation techniques. One assumption that can be made to simplify the equation is that the concentration of Cu^2+ ions (x) will be much smaller than the initial concentration of NH3. In that case, we can approximate the equation as: \(K_{value} = \frac{x}{0.10^4}\) Now, we can solve for x: \(x = K_{value} \times 0.10^4\) Once the value of x is calculated, it represents the concentration of Cu^2+ ions in the solution.

Key concepts

Equilibrium Expressions

Understanding chemical equilibria is key to grasping how chemical reactions balance out over time. An equilibrium expression is a way to quantify the balance of a reversible chemical reaction. It involves the concentrations of products over the concentrations of reactants, each raised to the power of their stoichiometric coefficients. Given a reaction like

\[aA + bB \rightleftharpoons cC + dD\],

the equilibrium expression for the reaction’s constant (\(K\)) is written as

\[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\].

In the case of the complex formation of copper(II) with ammonia, the equilibrium expression is represented by the formation constant (\(K_f\)), which specifically describes the concentration of the complex ion in relation to the concentrations of the individual ions that combine to form it. It’s important to note that pure solids and liquids are not included in equilibrium expressions, only species in the gaseous or aqueous phase are considered. When dealing with equilibrium problems, ensure all the concentration terms represent the same point in time at equilibrium. This is essential for accurate calculations and understanding of the system's dynamics.

ICE Table Method

When solving for unknowns in equilibrium problems, the ICE table method provides a structured approach to keep track of concentration changes. ICE stands for Initial, Change, and Equilibrium, representing different stages of the reaction.

The starting point is to note the initial concentrations. As a reaction proceeds, the concentrations change, generally modifying by an unknown amount we usually denote as \(x\). Then, by considering the stoichiometry of the balanced chemical equation, we can express the change for each reactant and product. This approach allows us to lay out the concentrations at equilibrium in a straightforward manner. For example, if we start with an initial concentration \([A]\) of a reactant \(A\), and it undergoes a decrease of \(x\) to form products, the equilibrium concentration would be \([A] - x\).

In our textbook exercise, the ICE table sets the stage for understanding how the presence of a complexing agent like ammonia affects the concentration of \(Cu^{2+}\) ions. The initial molarities, the expected changes (denoted by \(x\)), and the equilibrium concentrations provide the complete picture of the reaction’s progress. This clarity is essential for students as it helps in visualizing each step and ultimately finding the desired equilibrium concentration of \(Cu^{2+}\) ions.

Formation Constant (\(K_f\))

A specific type of equilibrium constant is the formation constant (\(K_f\)), which applies to the creation of complex ions in solution. It is indicative of the stability of the complex ion; higher \(K_f\) values suggest a more stable complex ion. In our example, the formation constant concerns the reaction where \(Cu^{2+}\) ions bond with ammonia (\(NH_3\)) to form the complex ion \(Cu(NH_3)_4^{2+}\).

The value of the formation constant plays a vital role in the predictive capacity of chemical equilibrium. It allows us to understand how much of the complex ion is formed under certain conditions. Students should know that \(K_f\) values are typically determined experimentally and are provided in reference materials, as they are a measure of the overall tendency of the complex ion to form. Remember that these constants are temperature-dependent, so always use the \(K_f\) value that corresponds to the conditions of your experiment or problem. Having a solid grasp on formation constants equips students with the ability to determine concentrations of species in equilibrium situations effectively.