A sample of hard water contains 20 mg of `Ca^(2+)` ions per litre. How many millimoles of `Na_(2)CO_(3)` would be requried to soften 1 L of the sample. Also calculate the mass of `Na_(2)CO_(3)`.

Options

A. 1
B. `10^(-3)`
C. 10
D. `10^(-5)`

Answer

To solve the problem step by step, we need to determine how many millimoles of sodium carbonate (Na₂CO₃) are required to soften 1 liter of hard water containing 20 mg of calcium ions (Ca²⁺). We will also calculate the mass of Na₂CO₃ needed.

### Step 1: Convert the mass of Ca²⁺ ions to grams
Given that the concentration of Ca²⁺ ions is 20 mg per liter, we convert this to grams:
\[ 20 \text{ mg} = 20 \times 10^{-3} \text{ g} = 0.020 \text{ g} \]

**Hint:** Remember that 1 mg = \(10^{-3}\) g.

### Step 2: Calculate the number of moles of Ca²⁺ ions
The molar mass of calcium (Ca) is approximately 40 g/mol. We can calculate the number of moles of Ca²⁺ ions using the formula:
\[ \text{Moles of Ca}^{2+} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
\[ \text{Moles of Ca}^{2+} = \frac{0.020 \text{ g}}{40 \text{ g/mol}} = 0.0005 \text{ mol} \]

**Hint:** Use the formula for moles, which is mass divided by molar mass.

### Step 3: Determine the stoichiometry of the reaction
The reaction between sodium carbonate (Na₂CO₃) and calcium ions (Ca²⁺) can be represented as:
\[ \text{Ca}^{2+} + \text{Na}_2\text{CO}_3 \rightarrow \text{CaCO}_3 + 2\text{Na}^{+} \]
From the balanced equation, we see that 1 mole of Na₂CO₃ reacts with 1 mole of Ca²⁺.

**Hint:** Identify the stoichiometric coefficients from the balanced chemical equation.

### Step 4: Calculate the number of moles of Na₂CO₃ required
Since the ratio of Ca²⁺ to Na₂CO₃ is 1:1, the number of moles of Na₂CO₃ required is equal to the number of moles of Ca²⁺:
\[ \text{Moles of Na}_2\text{CO}_3 = 0.0005 \text{ mol} \]

**Hint:** Use the stoichiometric ratio from the balanced equation to find the required moles.

### Step 5: Convert moles of Na₂CO₃ to millimoles
To convert moles to millimoles, we multiply by 1000:
\[ \text{Millimoles of Na}_2\text{CO}_3 = 0.0005 \text{ mol} \times 1000 = 0.5 \text{ mmol} \]

**Hint:** Remember that 1 mol = 1000 mmol.

### Step 6: Calculate the mass of Na₂CO₃ required
The molar mass of sodium carbonate (Na₂CO₃) is calculated as follows:
\[ \text{Molar mass of Na}_2\text{CO}_3 = (2 \times 23) + 12 + (3 \times 16) = 46 + 12 + 48 = 106 \text{ g/mol} \]
Now, we can calculate the mass of Na₂CO₃ required using the formula:
\[ \text{Mass of Na}_2\text{CO}_3 = \text{moles} \times \text{molar mass} \]
\[ \text{Mass of Na}_2\text{CO}_3 = 0.0005 \text{ mol} \times 106 \text{ g/mol} = 0.053 \text{ g} = 53 \text{ mg} \]

**Hint:** Use the formula for mass, which is moles multiplied by molar mass.

### Final Results
- Millimoles of Na₂CO₃ required: **0.5 mmol**
- Mass of Na₂CO₃ required: **53 mg**