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Derive Henderson - HasselBalch equation and mention its significance?

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Last updated date: 20th Jun 2024
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Answer
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Hint: It is an equation used in chemistry and also biochemistry. This equation is related to the pH value of a type of solution. If you know how it is related to pH value and for which type of solution, you will get a clear idea about what is Henderson – Hassel Balch equation.

Complete step by step answer:
Henderson – Hassel Balch equation:
It is used to estimate the pH value of a buffer solution. It provides a relationship between the pH of acids and there \[{\text{p}}{{\text{K}}_{\text{a}}}\].
Formula for Henderson – Hassel Balch equation:
\[{\text{pH = p}}{{\text{K}}_{\text{a}}} + {\text{ lo}}{{\text{g}}_{10}}(\dfrac{{{\text{[}}{{\text{A}}^ - }]}}{{{\text{[HA]}}}})\]
Here,
[\[{{\text{A}}^ - }\]] denotes the molar concentration of conjugate base
[HA] denotes the molar concentration of weak acid
So, formula for Henderson – Hassel Balch equation:
\[{\text{pH = p}}{{\text{K}}_{\text{a}}} + {\text{ lo}}{{\text{g}}_{10}}(\dfrac{{{\text{[conjugate base}}]}}{{{\text{[acid]}}}})\]
Derivation of Henderson – Hassel Balch equation:
\[
  {{\text{K}}_{\text{a}}}{\text{ = }}\dfrac{{[{{\text{H}}^ + }{\text{][}}{{\text{A}}^ - }{\text{]}}}}{{[{\text{HA]}}}} \\
  {\text{ - }}\log {\text{ }}{{\text{k}}_{{\text{a }}}} = {\text{ - log}}\dfrac{{[{{\text{H}}^ + }{\text{][}}{{\text{A}}^ - }{\text{]}}}}{{[{\text{HA]}}}} \\
  {\text{ - }}\log {\text{ }}{{\text{k}}_{{\text{a }}}}{\text{ = - log[}}{{\text{H}}^ + }]{\text{ - log}}\dfrac{{{\text{[}}{{\text{A}}^ - }{\text{]}}}}{{[{\text{HA]}}}} \\
\]
We know that,
\[
  {\text{ - log[}}{{\text{H}}^ + }]{\text{ = pH}} \\
  {\text{ - log[}}{{\text{K}}_{\text{a}}}]{\text{ = p}}{{\text{K}}_{\text{a}}} \\
\]
Rearranging the above equation:
\[{\text{pH = p}}{{\text{K}}_{\text{a}}} + {\text{log}}\dfrac{{[{\text{A]}}}}{{[{\text{HA]}}}}\]
This is the derivation of Henderson – Hassel Balch equation.
Significance of Henderson – Hassel Balch equation:
It is used to calculate the concentration of acid and salt when information of pH and total concentration of buffer solution is given.
It is used to calculate the pH of buffer solution.
Buffer solution can be prepared by adjusting the concentrations of the salt and acid that is added to the buffer. That buffer solution can be prepared for our desired pH.

Note: We can describe the buffer capacity of buffer solution. We can calculate the ionized and unionized concentrations of given chemicals. This equation has some limitations too. The main limitation is it has the assumption that the concentration of and its conjugate base will remain the same during the equilibrium.