A mixture contains 1.0 mole each of NaOH,`Na_(2)``CO_(3)` and `NaHCO_(3)`. When half of mixture is titrated with HCl ,it required x mole of HCl in presence of phenolphthalein. In another experiment ,half of mixture required y mole of same HCl in presence of methyl orange. Find the value of (x+y).

Answer

To solve the problem, we need to analyze the titration of a mixture containing 1 mole each of NaOH, Na2CO3, and NaHCO3 with HCl. We will consider two cases: one with phenolphthalein and the other with methyl orange.

### Step-by-Step Solution:

**Step 1: Understanding the Mixture**
- The mixture contains:
- 1 mole of NaOH
- 1 mole of Na2CO3
- 1 mole of NaHCO3
- Total moles in the mixture = 3 moles.

**Step 2: Titration with Phenolphthalein (Case 1)**
- When half of the mixture (1.5 moles total) is titrated with HCl in the presence of phenolphthalein, only NaOH and Na2CO3 will react.
- The reaction is:
- NaOH + HCl → NaCl + H2O (1:1 ratio)
- Na2CO3 + 2HCl → 2NaCl + H2O + CO2 (2:1 ratio)

- In half the mixture:
- Moles of NaOH = 0.5
- Moles of Na2CO3 = 0.5
- Therefore, the equivalents of NaOH reacting with HCl:
- For NaOH: 0.5 moles of NaOH will require 0.5 moles of HCl.
- For Na2CO3: 0.5 moles of Na2CO3 will require 0.5 * 2 = 1 mole of HCl.

- Total HCl required (x) in this case:
\[ x = 0.5 + 1 = 1.5 \text{ moles} \]

**Step 3: Titration with Methyl Orange (Case 2)**
- In the second case, when half of the mixture is titrated with HCl in the presence of methyl orange, all three components will react:
- NaOH reacts with HCl (1:1 ratio)
- Na2CO3 reacts with HCl (2:1 ratio)
- NaHCO3 reacts with HCl (1:1 ratio)

- In half the mixture:
- Moles of NaOH = 0.5
- Moles of Na2CO3 = 0.5
- Moles of NaHCO3 = 0.5

- Therefore, the equivalents of HCl required:
- For NaOH: 0.5 moles of NaOH will require 0.5 moles of HCl.
- For Na2CO3: 0.5 moles of Na2CO3 will require 0.5 * 2 = 1 mole of HCl.
- For NaHCO3: 0.5 moles of NaHCO3 will require 0.5 moles of HCl.

- Total HCl required (y) in this case:
\[ y = 0.5 + 1 + 0.5 = 2 \text{ moles} \]

**Step 4: Finding x + y**
- Now we can find the total:
\[ x + y = 1.5 + 2 = 3.5 \]

### Final Answer:
The value of \( x + y \) is **3.5**.

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