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I know sulfuric acid is a strong acid and acetic acid is a weak acid. However when I write them in their chemical symbol.

Sulfuric acid: $\ce{H2SO4}$

Acetic acid: $\ce{CH3COOH}$

I realise that acetic acid has 4 atoms of hydrogen whereas sulfuric acid has only 2 atoms of hydrogen. So, shouldn't acetic acid be stronger than sulfuric acid?

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Looking at the structures of those two acids might help. In both cases, the acidic hydrogens are part of hydroxyl groups. Because of polarity and the inductive effect, the O-H bond is weaker than C-H bonds which are very strong. As a result, it's nearly impossible for Carbon to lose a hydrogen without some kind of elimination reaction, whereas the O-H bond can autoprotolyze with other O-H bonds (even water). The weakness of the O-H bond is the basis for the acidity of those hydrogens. C-H bonds are essentially non-acidic, having pKa values around 45-50.

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The pKa of an acid tells you how strong the acid is.

$pKa=-log([H^+][A^-]/[HA])=-log(Ka)$ where $A$ is the conjugate base of the acid (the species left when the proton dissociates). A low value for $pKa$ gives the stronger acid since a high value for $Ka$ indicates that the equilibrium $\ce{HA<=>H+ + A-}$ lies well to the right.

There are a number of factors that make an acid's equilibrium position lie far to the right. By far the most important is the stability of the conjugate base ($A^-$). In your example, the reason $\ce{H2SO4}$ so much more acidic than $\ce{CH3COOH}$ because is can form more resonance structures than ethanoic acid so it distributes the negative charge better. Also the inductive effect of the alkyl group destabilizes the negative conjugate base. There are more reasons but this is the most important.

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Sulfuric acid is a strong acid; when placed in water, it completely dissociates into $H^+$ and $HSO_4^-$. While acetic acid dissociates into $H^+$ and $CH_3COO^-$; but being a weak acid, some of the molecules remain as $CH_3COOH$.

The difference between a strong and a weak acid is not necessarily their level of acidity; it is the degree to which they dissociate. Also as the previous answerer said, due to the nature of the bond of the 3 hydrogen molecules in acetate, we wouldn't consider it an acid because they don't break into $H^+$ ions as easily.

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