Determine [Zn2 ], [CN–], and [HCN] in a saturated solution of Zn(CN)2 with a fixed pH of 1.78. The Ksp for Zn(CN)2 is 3.0 × 10–16. The Ka for HCN is 6.2 × 10–10.

I know the following is what needs to be done, but im unsure of how to go about it.

1. Write the pertinent reactions. 2. Write the charge balance equation. 3. Write the mass balance equations. 4. Write the equilibrium constant expressions for each reaction. 5. Count the equations and unknowns. 6. Solve for all unknowns.

To determine the concentrations of [Zn2+], [CN-], and [HCN] in the saturated solution of Zn(CN)2 with a fixed pH of 1.78, we can follow the steps you mentioned:

1. Write the pertinent reactions:
a. Zn(CN)2 ⇌ Zn2+ + 2CN-
b. HCN ⇌ H+ + CN-

2. Write the charge balance equation:
[H+] + [Zn2+] = [CN-] + [HCN]

3. Write the mass balance equations:
For the first reaction:
[Zn(CN)2] = [Zn2+] + 2[CN-]
For the second reaction:
[HCN] = [H+] + [CN-]

4. Write the equilibrium constant expressions for each reaction:
For the first reaction: Ksp = [Zn2+][CN-]^2
For the second reaction: Ka = [H+][CN-]/[HCN]

5. Count the equations and unknowns:
We have 4 equations and 4 unknowns: [Zn2+], [CN-], [HCN], and [H+].

6. Solve for all unknowns:
Since the pH is given as 1.78, we know that [H+] = 10^(-pH) = 10^(-1.78) M.

Substituting this value into the charge balance equation, we have:
[CN-] + [Zn2+] = [HCN] + 10^(-1.78)

Now we can use the equilibrium constant expressions to relate the concentrations:
From the first reaction, we have:
[Zn2+] = (√(Ksp /[CN-]^2))

From the second reaction, we have:
[HCN] = ([H+][CN-])/Ka

Now we can substitute these expressions for [Zn2+] and [HCN] into the charge balance equation and obtain an equation involving only [CN-]:
[CN-] + (√(Ksp /[CN-])) = ([H+][CN-])/Ka + 10^(-1.78)

This equation can be solved numerically to find the concentration of [CN-]. Once you have [CN-], you can substitute it back into the expression for [Zn2+] and solve for [Zn2+]. Similarly, substitute [CN-] into the expression for [HCN] to find [HCN].

Remember to use appropriate units and consider the significance of any assumptions made in the calculations.