Calculate Unknown Mass X in Organic Combustion Analysis

To calculate the mass $X$ of the organic compound, we can use the percentage composition of carbon obtained from the combustion analysis.

Step 1: Calculate the Mass of Carbon in the Product

The molar mass of $\text{CO}_2$ is $12 + 2 \times 16 = 44\text{ g mol}^{-1}$.

The mass of carbon present in $0.25\text{ g}$ of $\text{CO}_2$ is given by: $\text{Mass of C} = \frac{\text{Molar mass of C}}{\text{Molar mass of CO}_2} \times \text{Mass of CO}_2$ $\text{Mass of C} = \frac{12}{44} \times 0.25\text{ g} = \frac{3}{44}\text{ g}$

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Step 2: Determine $X$ from the Percentage of Carbon

The percentage of carbon in the organic compound is given as $25\%$. By definition: $\% \text{ C} = \frac{\text{Mass of C}}{\text{Mass of organic compound } (X)} \times 100$

Substituting the known values: $25 = \frac{\frac{3}{44}}{X} \times 100$

Simplifying the equation: $\frac{25}{100} = \frac{3}{44 X}$ $\frac{1}{4} = \frac{3}{44 X}$ $44 X = 12$ $X = \frac{12}{44} = \frac{3}{11}\text{ g}$

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Step 3: Verification using Hydrogen Percentage

The molar mass of $\text{H}_2\text{O}$ is $2 \times 1 + 16 = 18\text{ g mol}^{-1}$.

The mass of hydrogen in $0.12\text{ g}$ of $\text{H}_2\text{O}$ is: $\text{Mass of H} = \frac{2}{18} \times 0.12\text{ g} = \frac{0.12}{9}\text{ g}$

Calculating the percentage of hydrogen: $\% \text{ H} = \frac{\frac{0.12}{9}}{\frac{3}{11}} \times 100 = \frac{0.12 \times 11 \times 100}{27} = \frac{132}{27} \approx 4.889\% \approx 4.89\%$ This matches the percentage given in the question.

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Step 4: Convert $X$ into $10^{-3}\text{ g}$ and Round

$X = \frac{3}{11}\text{ g} \approx 0.272727\text{ g}$

Expressing $X$ in terms of $10^{-3}\text{ g}$: $X = 272.727 \times 10^{-3}\text{ g}$

Rounding to the nearest integer yields: $X \approx 273 \times 10^{-3}\text{ g}$

Therefore, the correct option is A.