A 10 g sample of a mixture of calcium chloride and sodium chloride is treated with `Na_(2)CO_(3)` to precipitate calcium as calcium carbonate. This `CaCO_(3)` is heated to convert all the calcium to CaO and the final mass of CaO is 1.62 g. Calculate % by mass of NaCl in original solution.

Answer

To solve the problem, we need to determine the percentage by mass of sodium chloride (NaCl) in the original mixture of calcium chloride (CaCl₂) and sodium chloride (NaCl). We will follow these steps:

### Step 1: Calculate the moles of CaO produced
We are given that the mass of CaO produced is 1.62 g. First, we need to calculate the number of moles of CaO.

The molar mass of CaO (Calcium Oxide) is:
- Ca: 40.08 g/mol
- O: 16.00 g/mol
- Molar mass of CaO = 40.08 + 16.00 = 56.08 g/mol

Now, we can calculate the moles of CaO:
\[ \text{Number of moles of CaO} = \frac{\text{mass of CaO}}{\text{molar mass of CaO}} = \frac{1.62 \text{ g}}{56.08 \text{ g/mol}} \approx 0.0289 \text{ moles} \]

### Step 2: Relate moles of CaO to moles of CaCO₃
From the reaction, we know that 1 mole of CaCO₃ produces 1 mole of CaO. Therefore, the number of moles of CaCO₃ is also 0.0289 moles.

### Step 3: Calculate the mass of CaCO₃
Next, we need to find the mass of CaCO₃ produced. The molar mass of CaCO₃ (Calcium Carbonate) is:
- Ca: 40.08 g/mol
- C: 12.01 g/mol
- O: 16.00 g/mol (3 O atoms)
- Molar mass of CaCO₃ = 40.08 + 12.01 + (3 \times 16.00) = 100.09 g/mol

Now, we can calculate the mass of CaCO₃:
\[ \text{Mass of CaCO₃} = \text{Number of moles of CaCO₃} \times \text{Molar mass of CaCO₃} = 0.0289 \text{ moles} \times 100.09 \text{ g/mol} \approx 2.89 \text{ g} \]

### Step 4: Calculate the mass of CaCl₂ in the mixture
From the reaction:
\[ \text{CaCl₂} + \text{Na₂CO₃} \rightarrow \text{CaCO₃} + 2 \text{NaCl} \]
We see that 1 mole of CaCl₂ reacts to produce 1 mole of CaCO₃. Therefore, the mass of CaCl₂ that reacted can be calculated as follows:

Let the mass of CaCl₂ be \(y\) grams. The molar mass of CaCl₂ is:
- Ca: 40.08 g/mol
- Cl: 35.45 g/mol (2 Cl atoms)
- Molar mass of CaCl₂ = 40.08 + (2 \times 35.45) = 111.98 g/mol

Using the moles of CaCO₃ (which equals the moles of CaCl₂), we can find the mass of CaCl₂:
\[ y = \text{Number of moles of CaCl₂} \times \text{Molar mass of CaCl₂} = 0.0289 \text{ moles} \times 111.98 \text{ g/mol} \approx 3.23 \text{ g} \]

### Step 5: Calculate the mass of NaCl in the mixture
The total mass of the mixture is 10 g. The mass of NaCl can be calculated as:
\[ \text{Mass of NaCl} = \text{Total mass} - \text{Mass of CaCl₂} = 10 \text{ g} - 3.23 \text{ g} \approx 6.77 \text{ g} \]

### Step 6: Calculate the percentage by mass of NaCl
Finally, we can calculate the percentage by mass of NaCl in the original solution:
\[ \text{Percentage of NaCl} = \left( \frac{\text{Mass of NaCl}}{\text{Total mass}} \right) \times 100 = \left( \frac{6.77 \text{ g}}{10 \text{ g}} \right) \times 100 \approx 67.7\% \]

### Final Answer
The percentage by mass of NaCl in the original solution is approximately **67.7%**.