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I’m confused because in 3,5-dinitrophenol, the inductive effect is dominating whereas in 4-nitrophenol the mesomeric effect dominates. So which one should we consider for acidity?

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    $\begingroup$ Part of the answer depends on why you're asking. If you want to know the answer for which is acidic, you just look it up or measure it. You can't always determine the relative magnitude of competing effects in chemistry (unless you've got amazing intuition and/or experience), so you might not be able to determine the answer just by thinking about it. $\endgroup$ – Zhe Mar 1 at 14:36
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As Zhe's comment points out, intuition does not count for much here, unless you have enough actual experience working with these systems to know what is more acidic. There are some things which cannot be predicted a priori using simplified models.

The only way to find out is to look up the data. The acidity constants of some of these phenols are tabulated in J. Org. Chem. 1998, 63 (19), 6680–6688. 4-Nitrophenol has a $\mathrm pK_\mathrm a$ of $7.14$, and 3,5-dinitrophenol a $\mathrm pK_\mathrm a$ of $6.73$, in aqueous solution.

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I have used MarvinSketch 17.28 to estimate the acidic strengths of substituted nitrophenols, as measured by their $\mathrm pK_\mathrm a$ values.

$$\begin{array}{cc} \hline \text{Substitution} & \mathrm pK_\mathrm a \\ \hline \text{2,4,6-trinitro} & 2.344 \\ \text{2,4,5-trinitro} & 3.224 \\ \text{3,5-dinitro} & 6.238 \\ \text{4-nitro} & 7.075 \\ \text{3-nitro} & 7.892 \\ \hline \end{array}$$

It is evident that 3,5-dinitrophenol is more acidic than 4-nitrophenol.

The relative acidities of the meta- and para-monosubstituted nitrophenols are in line with predictions based on resonance arguments.

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    $\begingroup$ MarvinSketch seems to get most of those pKas fairly accurately, but I'm surprised at how bad the number for picric acid is. $\endgroup$ – orthocresol Mar 2 at 16:56

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