VCAA 2012 Q3
c) - Wouldn't the HCl also react with H2O, reducing its concentration, instead of just OH-?
d) - Why is the percentage dissociation of the acid to two significant figures instead of three (ie. 50% instead of 49.9%)?
When we convert pH to H3O+ concentration should we write that as
[H3O+] = 1.00x10-pH or [H3O+] = 10-pH?
For this question we couldn't assume that the initial concentration of the acid is equal to the equilibrium concentration.... so for what Ka values can we assume that [Acid]initially = [Acid]equilibrium and for what values can't we?
For the first point, water is the solvent. As the number of moles of water drop, so does the volume of water. Also, the concentration of water in water is so ridiculously high in comparison that you can neglect water in aqueous equilibria generally.
There are two factors at play here. Firstly, the Ka value needs to be sufficiently high. Secondly, the concentration can't be too low, otherwise the percentage ionisation will be too high.
I'll give an example. Let's assume that we have a solution of concentration M molar, acidity constant K.
Assume that the concentration of H+ and conjugate base formed is x.
So M-x molar acid left.
Subbing into K expression
K = x^2/(M-x)
x^2 = MK - xK
x^2 + xK - MK = 0
x = 1/2*(-K + sqrt(K^2+4MK)) quadratic formula, plus sign because x > 0
We want the case when K << M as that is what normally happens
x = 1/2*(-K + 2*sqrt(MK)*sqrt(1 + K/4M)) by bringing out the sqrt(4MK) from the square root
People are going to hate me for doing this, but I'm going to go ahead anyway.
(1+a)^n, for small a, is roughly equal to 1 + an + n(n-1)/2 * a^2
So apply this to the square root term
sqrt(1+K/4M) = (1+K/4M)^1/2, and assuming K/4M is small, which it generally is:
sqrt(1+K/4M) ~ 1+K/8M + 1/2*-1/2*(K/4M)^2 = 1+K/8M - K^2/64M^2
Subbing this back into the original expression for x
x~1/2*(-K+2sqrt(MK)*(1+K/8M - K^2/64M^2))
OK. Let's see what this means. If M is sufficiently larger than K, then K^2/64M^2 is meaninglessly small. We can discard that term.
If K/8M is also sufficiently small, then we can remove that term as well.
We are then left with x~1/2*(-K + 2sqrt(MK) = -K/2 + sqrt(MK)
The term sqrt(MK) is what you would normally get by assuming M>>x in K = x^2/(M-x)
But if you assumed that K/8M was tiny, you wouldn't care about the individual value of K, so the -K/2 term is also negligible in this case.
What does all this mean? You need to look at the ratio K/M and consider how significant it is to the concentration in M of acid produced. Generally, if you have a concentration of around 0.1 M, and you have a normal weak acid with a Ka of 10^-5, then K/M = roughly 10^-4, which is tiny. But if you had a solution of 10^-3 M and your Ka was 10^-4...then K/M is 0.1, which can't be neglected.