Calculate the equilibrium concentration of NH3,Cu^2+,[ Cu(NH3)]^2+ ,[Cu(NH3)2]^2+,[Cu(NH3)3]^2+,and[Cu(NH3)4]^2+ in a solution made by mixing 500.0 ml of 3.00 M NH3 with 500.0 ml of 2.00*10^-3 M Cu(NO3)2. The sequential equilibria are
Cu^2+ +NH3---->[Cu(NH3)]^2+ ka1=1.86x10^4
[Cu(NH3)]^2++NH3----->[Cu(NH3)2]^2+ k2= 3.88x10^3
[Cu(NH3)2]^2+ +NH3----->[Cu(NH3)3]^2+ k3=1.00x10^3
[Cu(NH3)3]^2+ +NH3----->[Cu(NH3)4]^2+ k4=1.55x10^2
To calculate the equilibrium concentrations of NH3, Cu^2+, [Cu(NH3)]^2+, [Cu(NH3)2]^2+, [Cu(NH3)3]^2+, and [Cu(NH3)4]^2+ in the solution, we will use the principles of sequential equilibria and the given equilibrium constants (ka1, k2, k3, k4).
First, let's determine the initial moles of NH3 and Cu(NO3)2 in the solution.
Moles of NH3:
Given volume = 500.0 ml = 0.5 L
Given concentration = 3.00 M
Moles of NH3 = Volume x Concentration = 0.5 L x 3.00 M = 1.50 moles
Moles of Cu(NO3)2:
Given volume = 500.0 ml = 0.5 L
Given concentration = 2.00*10^-3 M
Moles of Cu(NO3)2 = Volume x Concentration = 0.5 L x (2.00*10^-3 M) = 1.00*10^-3 moles
Now, we can use these initial moles to calculate the changes in moles for each sequential equilibrium.
For the first equilibrium:
Cu^2+ + NH3 -> [Cu(NH3)]^2+
Initial moles of Cu^2+ = 1.00*10^-3 moles (from Cu(NO3)2)
Initial moles of NH3 = 1.50 moles (from NH3)
Assuming x moles of Cu^2+ react with x moles of NH3, the change in moles will be -x for Cu^2+ and -x for NH3.
The equilibrium moles of Cu^2+ and [Cu(NH3)]^2+ will be (1.00*10^-3 - x) and x, respectively.
Using the equilibrium constant (ka1):
ka1 = ([Cu(NH3)]^2+) / ([Cu^2+][NH3])
1.86x10^4 = x / ((1.00*10^-3 - x) * (1.50 - x))
We can solve this equation to find the value of x, which represents the moles of [Cu(NH3)]^2+ and NH3 that reacted.
Similarly, repeat the above steps for the second, third, and fourth sequential equilibria using their respective equilibrium constants (k2, k3, k4).
Once you have calculated the values of x for each equilibrium, you can substitute the values into the expressions for the equilibrium moles of the respective species to determine their concentrations:
[NH3] = (Initial moles of NH3 - x) / Total solution volume
[Cu^2+] = (Initial moles of Cu^2+ - x) / Total solution volume
[Cu(NH3)]^2+ = x / Total solution volume
[Cu(NH3)2]^2+ = (Initial moles of [Cu(NH3)2]^2+ + x) / Total solution volume
[Cu(NH3)3]^2+ = (Initial moles of [Cu(NH3)3]^2+ + x) / Total solution volume
[Cu(NH3)4]^2+ = (Initial moles of [Cu(NH3)4]^2+ + x) / Total solution volume
Substitute the values you have calculated into the above expressions to find the equilibrium concentrations of NH3, Cu^2+, [Cu(NH3)]^2+, [Cu(NH3)2]^2+, [Cu(NH3)3]^2+, and [Cu(NH3)4]^2+ in the given solution.