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I'm supposed to explain why sulfuric acid and methanol are used in combination with 2,4-dinitrophenylhydrazine to form Brady's reagent. Are they just used as a solvent system for the hydrazine? Also, wouldn't sulfuric acid react with methanol?

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2,4-DNPH is usually dissolved in an alcoholic solvent, together with sulfuric acid, when used as a reagent to detect aldehydes and ketones (Brady's reagent). The sulfuric acid is a source of $\ce{H+}$, which catalyses the reaction between the hydrazine group and the carbonyl compound, as depicted in the mechanism below:

Mechanism of hydrazone formation

The final hydrazone product is the precipitate which signals a positive test.

The solvent used is typically methanol or ethanol. You are correct that these could theoretically react with sulfuric acid: dimethyl ether is the most likely product. However, the concentration of sulfuric acid is typically not sufficiently high, and these side reactions do not take place.

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2,4-Dinitrophenylhydrazine is available as a powder, synthesised from hydrazine sulphate and 2,4-dinitrochlorobenzene (Wikipedia).

Brady's reagent is prepared by dissolving the powder in a solution of methanol and concentrated sulfuric acid.

Although sulfuric acid and methanol react, the reaction is not 'big enough' to interfere with the 2,4-DNP reaction.

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