Convert the mass of $\mathrm{CO}_2$ to moles:

The molar mass of $\mathrm{CO}_2$ is 44 g/mol. Thus, the number of moles of $\mathrm{CO}_2$ produced is calculated as follows:

$ \mathrm{n}_{\mathrm{CO}_2} = \frac{220 \times 10^{-3} \, \text{g}}{44 \, \text{g/mol}} = 5 \times 10^{-3} \text{ moles} $

Calculate the mass of carbon in $\mathrm{CO}_2$:

Since each mole of $\mathrm{CO}_2$ contains one mole of carbon, the moles of carbon are the same: $5 \times 10^{-3}$ moles. The molar mass of carbon is 12 g/mol, so the mass of carbon is:

$ \mathrm{m}_{\mathrm{C}} = 5 \times 10^{-3} \, \text{moles} \times 12 \, \text{g/mol} = 60 \times 10^{-3} \, \text{g} = 60 \, \text{mg} $

Calculate the percentage composition of carbon:

To find the percentage by mass of carbon in the compound, use the formula:

$ \% \, \text{Carbon} = \left( \frac{60 \, \text{mg}}{500 \, \text{mg}} \right) \times 100 = 12\% $

Thus, the percentage composition of carbon in the compound is approximately 12%.