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I'm a little confused about what happens when you titrate a weak acid or base. For example, let's say that you are trying to titrate a weak acid. Based on this question, it seems like the weak acid isn't initially a buffer. But, as you begin to add strong base, conjugate base is created from the reaction between the weak acid and strong base. Does this mean that a buffer is created from the titration?

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I think that the above diagram will explain a lot of your questions.

it seems like the weak acid isn't initially a buffer

That is correct as to form a buffer you need a solution containing a weak acid and its conjugate weak base. So in the example above, initially the solution only contains acetic acid ($\ce{CH3COOH}$, but its conjugate base, acetate $\ce{CH3COO^{-}}$, is not present.

But, as you begin to add strong base, conjugate base is created from the reaction between the weak acid and strong base

That is absolutely correct. For example, in the above diagram the reaction that is occurring when you titrate the acetic acid with sodium hydroxide is: $$\ce{CH3COOH + NaOH -> CH3COO- + Na+ + H2O}$$

Now as you begin to titrate the acid, its conjugate base starts to form. This is supported by the fact that the pH doesn't really change when more $\ce{NaOH}$ is added to the solution.

To form an effective buffer solution you need roughly equal concentrations of the acid and conjugate base, hence that is why it says it is a buffer when half of the acid is titrated. However anywhere near that region can be considered as a buffer. Generally a solution is a buffer if the pH is within +/- 1 of its pKa. So for acetic acid, a buffer will have a pH between 3.76 and 5.76.

Does this mean that a buffer is created from the titration?

Yes, a buffer is created. However is doesn't stay as a buffer for long. As I mentioned above, when the ration of the concentration of acid and the conjugate base becomes too high or too low, it won't be a buffer. That is evident as when more $\ce{NaOH}$ is added to the buffer, the concentration of the conjugate base becomes much greater than the concentration of the acid, resulting in the pH to increase dramatically.

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