A mixture of `CaCO_(3)` and `MgCO_(3)` weighing `1.84 g` on heating left a residue weighing `0.96 g`. Calculate the percentage of each in the mixture.

Answer

To solve the problem of determining the percentage of calcium carbonate (CaCO₃) and magnesium carbonate (MgCO₃) in a mixture weighing 1.84 g that leaves a residue of 0.96 g upon heating, we can follow these steps:

### Step 1: Understand the Reaction
When CaCO₃ and MgCO₃ are heated, they decompose into their respective oxides and carbon dioxide gas is released:
- \( \text{CaCO}_3 \rightarrow \text{CaO} + \text{CO}_2 \)
- \( \text{MgCO}_3 \rightarrow \text{MgO} + \text{CO}_2 \)

The total mass of the residue after heating is the sum of the masses of CaO and MgO.

### Step 2: Define Variables
Let:
- \( x \) = mass of CaCO₃ in grams
- Therefore, the mass of MgCO₃ = \( 1.84 - x \) grams

### Step 3: Calculate Mass of Oxides Produced
From the decomposition reactions, we know:
- 100 g of CaCO₃ produces 56 g of CaO
- 84 g of MgCO₃ produces 36 g of MgO

Thus, the mass of CaO produced from \( x \) grams of CaCO₃ is:
\[ \text{Mass of CaO} = \frac{56}{100} \times x = 0.56x \text{ g} \]

The mass of MgO produced from \( 1.84 - x \) grams of MgCO₃ is:
\[ \text{Mass of MgO} = \frac{36}{84} \times (1.84 - x) = \frac{3}{7} \times (1.84 - x) \text{ g} \]

### Step 4: Set Up the Equation
The total mass of the residue (CaO + MgO) is given as 0.96 g:
\[ 0.56x + \frac{3}{7}(1.84 - x) = 0.96 \]

### Step 5: Solve the Equation
First, simplify the equation:
\[ 0.56x + \frac{3}{7} \times 1.84 - \frac{3}{7}x = 0.96 \]
Calculating \( \frac{3}{7} \times 1.84 \):
\[ \frac{3 \times 1.84}{7} = \frac{5.52}{7} \approx 0.78857 \]
Now substituting back into the equation:
\[ 0.56x - \frac{3}{7}x + 0.78857 = 0.96 \]
Convert \( \frac{3}{7} \) to a decimal:
\[ \frac{3}{7} \approx 0.42857 \]
So the equation becomes:
\[ (0.56 - 0.42857)x + 0.78857 = 0.96 \]
\[ 0.13143x + 0.78857 = 0.96 \]
Subtract \( 0.78857 \) from both sides:
\[ 0.13143x = 0.17143 \]
Now, divide by \( 0.13143 \):
\[ x \approx \frac{0.17143}{0.13143} \approx 1.30 \text{ g} \]

### Step 6: Calculate Mass of MgCO₃
Now, calculate the mass of MgCO₃:
\[ \text{Mass of MgCO}_3 = 1.84 - 1.30 = 0.54 \text{ g} \]

### Step 7: Calculate Percentages
Now we can find the percentages:
- Percentage of CaCO₃:
\[ \text{Percentage of CaCO}_3 = \left(\frac{1.30}{1.84}\right) \times 100 \approx 70.66\% \]
- Percentage of MgCO₃:
\[ \text{Percentage of MgCO}_3 = \left(\frac{0.54}{1.84}\right) \times 100 \approx 29.34\% \]

### Final Answer
- Percentage of CaCO₃ = 70.66%
- Percentage of MgCO₃ = 29.34%