To solve the problem, we will follow these steps:
### Step 1: Determine the mass of the first metal
Given that one metal is present to the extent of 12.5% by mass in the mineral, we can calculate the mass of this metal in the 2.8 g sample.
\[
\text{Mass of metal M} = \frac{12.5}{100} \times 2.8 \, \text{g} = 0.35 \, \text{g}
\]
### Step 2: Calculate the mass of carbonates lost
When the mineral is heated, it loses 1.32 g of carbon dioxide (CO₂). The reaction of the carbonates can be represented as follows:
\[
\text{MCO}_3 + \text{M'}\text{CO}_3 \rightarrow \text{MO} + \text{M'}\text{O} + \text{CO}_2
\]
From the stoichiometry of the reaction, we know that 1 mole of carbonate produces 1 mole of CO₂.
### Step 3: Calculate the mass of carbonate that produced the CO₂
The molar mass of carbonate (CO₃) is calculated as follows:
\[
\text{Molar mass of CO}_3 = 12 \, (\text{C}) + 3 \times 16 \, (\text{O}) = 60 \, \text{g/mol}
\]
The molar mass of CO₂ is:
\[
\text{Molar mass of CO}_2 = 12 \, (\text{C}) + 2 \times 16 \, (\text{O}) = 44 \, \text{g/mol}
\]
Using the ratio of the masses, we can find out how much carbonate corresponds to the 1.32 g of CO₂ produced:
\[
\text{Mass of carbonate} = \left(\frac{60 \, \text{g (carbonate)}}{44 \, \text{g (CO}_2)}\right) \times 1.32 \, \text{g (CO}_2) = 1.8 \, \text{g (carbonate)}
\]
### Step 4: Calculate the total mass of metals in the sample
The total mass of the sample is 2.8 g. The mass of the carbonates is 1.8 g, so the mass of the metals is:
\[
\text{Mass of metals} = \text{Total mass} - \text{Mass of carbonates} = 2.8 \, \text{g} - 1.8 \, \text{g} = 1.0 \, \text{g}
\]
### Step 5: Calculate the mass of the second metal
We already calculated the mass of the first metal (M) as 0.35 g. Therefore, the mass of the second metal (M') can be calculated as:
\[
\text{Mass of metal M'} = \text{Total mass of metals} - \text{Mass of metal M} = 1.0 \, \text{g} - 0.35 \, \text{g} = 0.65 \, \text{g}
\]
### Step 6: Calculate the percentage by mass of the second metal
Finally, we can calculate the percentage by mass of the second metal (M') in the mineral:
\[
\text{Percentage by mass of metal M'} = \left(\frac{\text{Mass of metal M'}}{\text{Total mass of sample}}\right) \times 100 = \left(\frac{0.65 \, \text{g}}{2.8 \, \text{g}}\right) \times 100 \approx 23.21\%
\]
### Final Answer
The percentage by mass of the other metal is approximately **23.21%**.
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