The problem is as follows:
The toxic metal cadmium Cd2+ has a tendency to complex with as many as 4 Cl- ions. The complexation reactions can be written as:
Cd2+ + Cl- ---> CdCl+ log Kf,1 = 1.98
Cd2+ + 2Cl- ---> CdCl2^0 log Kf,2 =2.60
Cd2+ + 3Cl- ---> CdCl3- log Kf,3 = 2.40
Cd2+ + 4Cl- ----> CdCl42- log Kf,4 = 2.50
Task: Compute the percentage of total Cd2+ in a solution that remains uncomplexed (free Cd2+ ) if the Cl- concentration is 0.005 M (ignore activity coefficient correction, assume activities equal concentrations).
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So I know that I will likely have to use the complex formation equilibrium constants in tandem with the initial Cl- concentration of 0.005 M, but I'm not exactly sure how to execute the problem. How can I obtain insight into the total uncomplexed Cd2+ with just this information? I was thinking about setting up each species with regard to the given K constants and plugging and iterating for values of Cd2+, but this doesn't seem like the route to go. Anybody know of any details I may need to make use of / where to start with this problem? Any help, in any form, would be appreciated.
To solve the problem, you need to consider the equilibrium reactions for each step of cadmium Cd2+ complexation with Cl- ions. From the given information, you know the equilibrium constant (Kf) values for each reaction.
Let's denote the concentration of free Cd2+ as [Cd2+], and the concentration of Cl- as [Cl-]. Initially, you have [Cl-] = 0.005 M.
The first step is to write the equations for the complexation reactions using the equilibrium constants:
Cd2+ + Cl- ---> CdCl+ (Kf,1 = 1.98)
Cd2+ + 2Cl- ---> CdCl2^0 (Kf,2 = 2.60)
Cd2+ + 3Cl- ---> CdCl3- (Kf,3 = 2.40)
Cd2+ + 4Cl- ---> CdCl42- (Kf,4 = 2.50)
Now, let's assume x represents the concentration of Cd2+ that remains uncomplexed. This means that the concentration of each complexed species can be expressed in terms of x:
[CdCl+] = x
[CdCl2^0] = Kf,1 * [Cd2+] * [Cl-] = 1.98 * x * [Cl-]
[CdCl3-] = Kf,1 * Kf,2 * [Cd2+] * [Cl-]^2 = 1.98 * 2.60 * x * [Cl-]^2
[CdCl42-] = Kf,1 * Kf,2 * Kf,3 * [Cd2+] * [Cl-]^3 = 1.98 * 2.60 * 2.40 * x * [Cl-]^3
The total concentration of Cd2+ can be written as:
[Cd2+] = [Cd2+]_initial - ([CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-])
Since we assume all Cl- ions come from CdClx complex formation, [Cl-] = [CdCl+] + 2[CdCl2^0] + 3[CdCl3-] + 4[CdCl42-].
To solve for x, you need to set up an equation and solve for x. First, substitute the expressions you derived for [Cd2+] and [Cl-] into the equation:
[Cd2+] = [Cd2+]_initial - ([CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-])
[Cl-] = [CdCl+] + 2[CdCl2^0] + 3[CdCl3-] + 4[CdCl42-]
Then, substitute [Cd2+]_initial = [Cd2+] + [CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-]:
[Cd2+]_initial - ([CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-]) = [Cd2+] + [CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-]
Simplify the equation:
[Cd2+]_initial - [Cd2+] = [CdCl+] + [CdCl2^0] + [CdCl3-] + [CdCl42-]
Now, substitute the expressions derived for the complexed species in terms of x:
[Cd2+]_initial - [Cd2+] = x + 1.98 * x * [Cl-] + 1.98 * 2.60 * x * [Cl-]^2 + 1.98 * 2.60 * 2.40 * x * [Cl-]^3
Finally, substitute [Cl-] = 0.005 M and solve for x. The value of x represents the percentage of total Cd2+ in the solution that remains uncomplexed.