The correct answer is Option B : Both S and E are true, and E is not the correct explanation of S.

Let us examine both the statement and the explanation :

STATEMENT (S) : In the titration of Na₂CO₃ with HCl using methyl orange indicator, the volume required at the equivalence point is twice that of the acid required using phenolphthalein indicator.

This statement is true. When titrating Na₂CO₃ (sodium carbonate), which is a diprotic base, there are two stages of the neutralization reaction. The first stage is the neutralization of carbonate ion (CO₃^2-) to bicarbonate ion (HCO₃^-), which is indicated by phenolphthalein changing color. Phenolphthalein changes color at a pH around 8.3, which corresponds to the first endpoint of the titration:

$$ \text{Na}_2\text{CO}_3 + \text{HCl} \rightarrow \text{NaHCO}_3 + \text{NaCl} $$

The second stage is the neutralization of bicarbonate ion to carbonic acid (H₂CO₃), which then decomposes into water and carbon dioxide. Methyl orange, which changes color at a pH of about 3.1 to 4.4, is used to indicate this second endpoint :

$$ \text{NaHCO}_3 + \text{HCl} \rightarrow \text{H}_2\text{CO}_3 + \text{NaCl} $$

$$ \text{H}_2\text{CO}_3 \rightarrow \text{H}_2\text{O} + \text{CO}_2 $$

Therefore, when using methyl orange, the total volume of HCl required for complete neutralization (both stages) is indeed twice the amount needed to reach the first endpoint with phenolphthalein.

EXPLANATION (E) : Two moles of HCl are required for the complete neutralization of one mole of Na₂CO₃.

This explanation is true. As evident from the chemical equations above, one mole of Na₂CO₃ requires one mole of HCl to convert to NaHCO₃, and then another mole of HCl to neutralize the NaHCO₃ further into H₂CO₃ followed by decomposition to water and CO₂. Therefore, the total is two moles of HCl per mole of Na₂CO₃.

Although both the statement and the explanation are true, the explanation does not adequately explain why the volume of HCl is doubled with methyl orange as the indicator compared to phenolphthalein. The correct explanation should involve the different end points detected by the two indicators (phenolphthalein indicating the first stage, methyl orange the second stage), rather than the overall stoichiometry of the reaction.