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For a titration, if you want to know the concentration of an acid you use a strong base such as $\ce{NaOH}$. My question is; why you can't use a weak base for this purpose? Is it because the calculation would be tougher? Is it because you'd need more of the base?

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There is no reason why you cannot use a weak base to titrate a strong acid. The calculation is probably considered a bit more complex but I think the real reason is a practical one. For the sake of simplicity, I swapped your question around just a bit, and the figure below shows the titration of a strong base (blue) weak base (reddish) and very weak base (dashed line) with a strong acid (say, for example 1 M $\ce{HCl}$).

simulated titration curves

In each case the concentration of the acid is the same (1.0 M) and the concentration of the base is 0.100 M. The volume of the base titrated is 100 mL. For the standard strong acid/strong base scenario, one finds a well defined equivalence point. When you transition to a weak base, however, there is additional information presented in the titration curve, namely the pKb of the base, which is a measure of its strength. The pKb of the base can be obtained by finding the pH at the half-way mark between no acid added and the equivalence point (and if you know a bit of calculus, this is the point of minimum slope prior to the equivalence point). I've marked the pKb of the reddish titration curve with a dot, and the pKb is 9. As the base gets weaker (less willing to accept a proton) the region before the equivalence point becomes less well defined. In an extreme example, I have shown a simulated titration curve of acetic acid (a very poor base indeed!) with a strong acid. Notice how the equivalence point in this curve would be very hard to pick out if we didn't know where we were looking.

So the bottom line is this: the calculations aren't harder (one uses the Henderson–Hasselbalch equation and this is a standard exercise in a first year analytical chemistry class) but the data one obtains from the experiment are harder to analyze. And in practice, we do perform weak/strong acid/base titrations when we are interested in determining the strength of the acid or base under consideration.

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Because a strong base should be able to titrate nearly any acid, strong or weak. Recall that strong/weak does not mean concentrated/dilute! It means that, for an acid, it is easy/hard for the acid to donate a proton to the base's hydroxide ion. If you use a weak base to titrate a weak acid, you may not be able to accomplish anything if the base is too weak, because the weak acid doesn't want to donate the proton, and the weak base doesn't want to accept it. Even if you use a strong base, it may be the case that your acid is so weak that even your strong base is not strong enough, but this is rare. This is where you use the so called $\ce{pK_a}$ values of the acid and base to determine if you can get a reaction, which is the titration.

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If you use strong base for titration the titration curve will be very steep. It makes it easier to observe the equilibrium point.

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