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How does a mixture of \[C{{H}_{3}}COOH\] and \[C{{H}_{3}}COONa\]act as a buffer when a small amount of \[NaOH\] is added?

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Answer
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Hint: We know that an acid buffer solution is a mixture of weak acid and its salt with strong base. A basic buffer solution is the mixture of weak base and its salt with strong acid. Similarly when there is addition of a small amount of strong base to a buffer solution to form conjugate base and water.

Complete answer:
Buffer solutions have the ability to resist the pH of the solution. When we add a small amount of weak acid or weak base, there is no change in the pH. When there is addition of a small amount of strong acid, the hydrogen ions from strong acid combine with the conjugate base of the buffer solution to form feebly ionized weak acid. Similarly when there is addition of a small amount of strong base to a buffer solution to form feebly ionized weak acid.
A mixture of a weak acid and its salt with a strong base serves as an acidic buffer: \[C{{H}_{3}}COO{{H}_{\left( aq \right)}}+{{H}_{2}}{{O}_{\left( l \right)}}\to CH3COO_{\left( aq \right)}^{-}+{{H}_{3}}O_{\left( aq \right)}^{+}.\] \[C{{H}_{3}}COOH\] is a weak acid, while is a weak base. So, if \[~NaOH\] is added to the system the following reaction occurs: \[C{{H}_{3}}COO{{H}_{\left( aq \right)}}+NaO{{H}_{\left( aq \right)}}\to CH3COO_{\left( aq \right)}^{-}+{{H}_{2}}{{O}_{\left( l \right)}}+Na_{\left( aq \right)}^{+}.\]
The reaction shows that some acetic acids react with the \[OH\] from the base and converts it to water. Therefore, the pH will not change as drastically as it would have without the buffer. Sodium acetate ionizes completely providing the acetate ion, \[C{{H}_{3}}CO{{O}^{}}.\]

Note:
Remember that a buffer solution resists change in pH. When a small amount of an acid (or a base) is added to the buffer solution, the pH of the solution remains unchanged. For example a mixture of weak acid (acetic acid) and its salt ammonium acetate with weak base (ammonia). Hence, it is not a buffer Solution.