To maintain the effectiveness of a buffer, relative concentrations of the acid and the conjugate base should not differ by a factor of 10. Based on this information, derive pH range within which the buffer can work effectively.
I know that I have to use the HH equation, but I cannot derive it.
To derive the Henderson-Hasselbalch equation, we need to first understand some concepts. The Henderson-Hasselbalch equation relates the pH of a buffer solution to the concentration of the acid and the conjugate base. It is derived from the equation for the ionization of an acid:
[H+][A-] / [HA] = Ka
where [H+] is the concentration of H+ ions, [A-] is the concentration of the conjugate base, [HA] is the concentration of the acid, and Ka is the acid dissociation constant.
To derive the Henderson-Hasselbalch equation, we want to express the equation above in terms of the logarithm of the concentrations, which would give us the pH. We can start by taking the logarithm of both sides:
log ([H+][A-] / [HA]) = log Ka
Using the logarithm property that states log (a / b) = log a - log b:
log [H+] + log [A-] - log [HA] = log Ka
Next, we can use the fact that pH is defined as the negative logarithm of the hydrogen ion concentration:
-pH + log [A-] - log [HA] = log Ka
Rearranging the equation and multiplying both sides by -1:
pH = -log Ka + log [A-] - log [HA]
Simplifying further, we get:
pH = pKa + log ([A-] / [HA])
This is the Henderson-Hasselbalch equation. It tells us that the pH of a buffer solution is determined by the pKa of the acid, the logarithm of the ratio of the conjugate base concentration to the acid concentration ([A-] / [HA]).
Now, let's determine the pH range within which the buffer can work effectively. According to the information given, the relative concentrations of the acid and the conjugate base should not differ by a factor of 10. This means that the ratio [A-] / [HA] should be within the range 0.1 to 10 (or 1/10 to 10/1, which is equivalent), as a difference of more than 10 would significantly impact the buffer's effectiveness.
Therefore, for a buffer to work effectively, the pH should be within the range calculated using the Henderson-Hasselbalch equation:
pH = pKa + log ([A-] / [HA])
Substituting the range 0.1 to 10 for ([A-] / [HA]):
pH = pKa + log (0.1) to (10)
This gives us the pH range within which the buffer can work effectively.