What fraction of piperazine (perhydro-1,4-diazine) is in each of its three forms (H2A, HA–, A2–) at pH 8.67? The acid dissociation constant values for piperazine are Ka1 = 4.65×10–6 and Ka2 = 1.86×10–10.
You do these using the alpha system.
The denominator of the fraction is
D = (H^+)^2 + k1*(H^+) + k1k1
alphao = (H2A) = (H^+)^2/D
alpha1 = (HA^-) = k1*(H^+)/D
alpha2 = (A^2-) = k1k2/D
Plug in the numbers and solve.
THIS IS WRONG. DR. BOB IS WRONG YET AGAIN. DON'T WASTE YOUR TIME LOOKING AT THIS.
Well, it seems like piperazine is having quite the identity crisis! Let's see if we can sort it out.
Since we're dealing with a two-step dissociation process, we'll need to do some math. At pH 8.67, which is relatively basic, we can assume that the majority of piperazine will be in its deprotonated forms.
Let's start with HA–. To find the fraction of piperazine in this form, we'll use the equation:
[H+] * [A-] / [HA] = Ka1
Since we know the value of Ka1 (4.65×10–6), we can rearrange the equation to solve for [A-] / [HA]. However, we don't have the concentrations of the species, so we'll have to use the Henderson-Hasselbalch equation:
pH = pKa + log([A-] / [HA])
By plugging in the values for pH (8.67) and pKa1 (-log(Ka1)), we can solve for [A-] / [HA]. This will give us the ratio of deprotonated piperazine to protonated piperazine.
Similarly, we can find the ratio of A2– to HA– using Ka2 and the Henderson-Hasselbalch equation.
As for H2A, which is the fully protonated form, we can assume that at pH 8.67, it will be present in negligible amounts.
So, to summarize in a slightly less serious way, piperazine is splitting its personality at pH 8.67, with most of it leaning towards the deprotonated side as HA– and A2–. H2A is feeling a bit left out, so it decides to be a loner.
Now, let's crunch those numbers and find the fractions!
To determine the fraction of piperazine in each of its three forms (H2A, HA–, A2–) at pH 8.67, we need to calculate the concentrations of each form.
First, let's define the terms:
- H2A represents the fully protonated form of piperazine.
- HA– represents the singly protonated form of piperazine.
- A2– represents the deprotonated form of piperazine.
Now, let's set up the equilibrium reactions and their respective equilibrium constants (Ka) for piperazine:
1) H2A ↔ HA– + H+
Ka1 = [HA–][H+] / [H2A]
2) HA– ↔ A2– + H+
Ka2 = [A2–][H+] / [HA–]
At the given pH of 8.67, we can assume that [H+] is equal to 10^(-pH).
To find the concentrations, we need to consider that the sum of all three forms of piperazine remains constant:
[H2A] + [HA–] + [A2–] = Total concentration of piperazine (which we can denote as [Pip])
Let's assume the total concentration of piperazine is 1M.
Now, we can solve these equations step by step:
Step 1: Calculate [H+]
[H+] = 10^(-8.67)
Step 2: Calculate [HA–] using Ka1
[H2A] = [Pip] * [H2A] / [H2A + HA– + A2–]
[HA–] = [Pip] * [H2A] / Ka1 * [H+]
Step 3: Calculate [A2–] using Ka2
[A2–] = [Pip] * [HA–] / Ka2 * [H+]
Now, we have the concentrations of each form of piperazine, and we can calculate the fraction of each form.
To find the fraction of each form, divide the concentration of each form by the total concentration:
Fraction of H2A = [H2A] / [Pip]
Fraction of HA– = [HA–] / [Pip]
Fraction of A2– = [A2–] / [Pip]
Let's plug in the numbers:
[H+] = 10^(-8.67)
[H2A] = [Pip] * [H2A] / [H2A + HA– + A2–]
[HA–] = [Pip] * [H2A] / Ka1 * [H+]
[A2–] = [Pip] * [HA–] / Ka2 * [H+]
Fraction of H2A = [H2A] / [Pip]
Fraction of HA– = [HA–] / [Pip]
Fraction of A2– = [A2–] / [Pip]
Now, you can substitute the given values for Ka1, Ka2, and pH, and perform the calculations to find the fractions of each form.