# Systematic treatment of equilibrium

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1. Jul 3, 2016

### reminiscent

1. The problem statement, all variables and given/known data
Determine [Zn2+], [CN–], and [HCN] in a saturated solution of Zn(CN)2 with a fixed pH of 1.060. The Ksp for Zn(CN)2 is 3.0 × 10–16. The Ka for HCN is 6.2 × 10–10.

2. Relevant equations

3. The attempt at a solution
I don't have a phone or camera right now to take a picture of my work, but here's my steps summarized:
1. I found that the [H+]= 0.0871 M.
2. I used that and the Kw equation to find [OH-] = 1.15 x 10^-13 M.
3. I used the mass balance equation to find a new equation for [HCN].
4. I plugged that into the Kb equation, along with the [OH-] in order to find [CN-].
5. I found [CN-] = 4.31 x 10^-24, then plugged that into the Ksp equation to find [Zn2+] = 1.61 x 10^31 and [HCN] = 3.22 x 10^31. The system said my answers were incorrect.
I tried another way using the ICE method and not using systematic treatment of equilibrium, and I found [Zn2+] = 0.2455 M, [CN-] = 0.491 M, and [HCN] = 68.98 x 10^6 M, but the system still said it was wrong.

Last edited: Jul 3, 2016
2. Jul 3, 2016

### Staff: Mentor

Why do you use Kb and [OH-] instead of just converting Kb to Ka?

How?

Or incorrect?

In general once you convert Kb to Ka you will end with three equations in three unknowns.

3. Jul 3, 2016

### reminiscent

Sorry, I will edit my post for certain mistakes...
They gave us the Ka for HCN is 6.2 × 10–10.
They converted it to Kb. I don't understand what you mean, though.
For the mass balance equation, I just did: [HCN] = 2[Zn2+] - [CN-].
I'm so stuck, I don't understand how to correctly find it.

4. Jul 3, 2016

### Staff: Mentor

If you are given Ka and pH, there is no need to go through the reaction between CN- and water,

HCN ↔ H+ +CN-

is perfectly enough to describe the HCN/CN- equilibrium.

Write Ka - this will be your first equation (with two unknowns, as pH is fixed).

Second equation is the mass balance (but it can be simplified - see below).

Third equation is Ksp.

That gives three equations in three unknowns, the rest is just a math. HCN is a very weak acid and it is protonated almost quantitatively at so low pH (so [HCN] >> [CN-] and the latter can be ignored in the mass balance).

5. Jul 3, 2016

### reminiscent

Okay, I understand - so the equations are:
[HCN] = 2[Zn2+]
Ka = [OH-][HCN]/[CN-] = 6.2 x 10^-10
Ksp = [Zn2+][CN-]^2 = 3.0 x 10^-16

Which one should I solve for first? I think the part of solving is what confuses me.

6. Jul 3, 2016

### reminiscent

Okay so far, I made the mass balance equation and Ksp equation equal to each other by isolating [Zn2+], and I solved for [HCN]. Can I just plug that back into the Ka equation, then solve for [CN-] this way? Also, [OH-] would be 1.15 x 10^-13 M, correct?

7. Jul 3, 2016

### reminiscent

I got [CN-] = 4.81 x 10-7, [Zn2+] = 0.00130, and [HCN] = 0.00259 and it is still incorrect...

8. Jul 4, 2016

### Staff: Mentor

I can confirm it is incorrect - but there is not much I can do not seeing what and how you did.

9. Jul 4, 2016

I worked this one and it is interesting. I think I got the correct answer. It uses 3 equations and 3 unknowns. Algebraically it is actually kind of simple. In post #5 your $K_a$ is incorrect and $[H+]$ belongs in the equation instead of $[OH-]$. Since you know the pH, you know the numerical value of $[H+]$. The algebra is simple enough that you can use your complete mass balance of equation of post #3. You can eliminate the $[HCN]$ from the 3 equations using the mass balance equation, and eliminate the $[Zn+2]$ as well and solve for $[CN-]$.

10. Jul 4, 2016

### reminiscent

I think I forgot my algebra skills because I got it wrong again...
Okay I tried what you did. I used the mass balance equation of post 3, plugged that into the Ka equation for [HCN], then I simplified more so I get Ka = 0.0871*2*([Zn2+]/[CN-]). I made the Ksp equation equal to [Zn2+], I plugged that into the Ka equation for [Zn2+]. The [CN-] would have a power of 3 to it, and I got [CN-]=0.00438. It is still wrong.

11. Jul 4, 2016

Your $K_a$ equation is incorrect. Your mass balance is $[CN-]=2[Zn+2]-[HCN]$ and the shortcut version is $[HCN]=2 [Zn+2]$. In any case $[CN-]$ should be in the numerator and doesn't get replaced. You could put either $[HCN]=(2[Zn+2])$ or $[HCN]=(2[Zn+2]-[CN-])$ in the denominator. ($[CN-]$ is very small, but you really don't need any shortcuts for the algebra.) Y(our number .0871 for the $[H+]$ for the pH =1.06 looks correct.) Your $K_a$ equation is also incorrect in post # 5. It is actually a simple equation and should read $K_a= ([H+][CN-])/[HCN]$. That appears where your error originated...

Last edited: Jul 4, 2016
12. Jul 4, 2016

I added a couple of things to post #11. Please be sure to read the latest edited version.

13. Jul 4, 2016

### reminiscent

Thank you so much! I can't believe I didn't catch that. I finally got it correct. Thank you!

14. Jul 4, 2016

### Staff: Mentor

In the future please show your work - it is much easier to find a mistake seeing what was done, than guessing out of rather cryptic description.

Plus, when you try to put your work in a readable format you are quite likely to find mistakes by yourself.