Calculating Hydrogen Percentage Composition in Ethane ($C_2H_6$)

This question asks us to find the percentage of hydrogen by mass in the compound ethane, which has the chemical formula $C_2H_6$. To determine this, we need to calculate the molar mass of ethane and the total mass contributed by hydrogen atoms within it.

Understanding Percentage Composition by Mass

The percentage composition by mass of an element in a compound tells us what fraction of the compound's total mass is contributed by that element. The formula used is:

$$ \text{Percentage by mass of element} = \left( \frac{\text{Total mass of the element in the compound}}{\text{Molar mass of the compound}} \right) \times 100\% $$

Step-by-Step Calculation for Ethane ($C_2H_6$)

1. Determine Atomic Masses:
   • The approximate atomic mass of Carbon (C) is $12.01$ atomic mass units (amu) or g/mol.
   • The approximate atomic mass of Hydrogen (H) is $1.008$ atomic mass units (amu) or g/mol.
2. Calculate the Molar Mass of Ethane ($C_2H_6$):

   Ethane has 2 carbon atoms and 6 hydrogen atoms. So, its molar mass is calculated as:

   $$ \text{Molar Mass of } C_2H_6 = (2 \times \text{Atomic mass of C}) + (6 \times \text{Atomic mass of H}) $$

   $$ \text{Molar Mass of } C_2H_6 = (2 \times 12.01 \text{ g/mol}) + (6 \times 1.008 \text{ g/mol}) $$

   $$ \text{Molar Mass of } C_2H_6 = 24.02 \text{ g/mol} + 6.048 \text{ g/mol} $$

   $$ \text{Molar Mass of } C_2H_6 = 30.068 \text{ g/mol} $$

3. Calculate the Total Mass of Hydrogen in Ethane:

   There are 6 hydrogen atoms in ethane ($C_2H_6$).

   $$ \text{Total Mass of H in } C_2H_6 = 6 \times \text{Atomic mass of H} $$

   $$ \text{Total Mass of H in } C_2H_6 = 6 \times 1.008 \text{ g/mol} $$

   $$ \text{Total Mass of H in } C_2H_6 = 6.048 \text{ g/mol} $$

4. Calculate the Percentage Composition of Hydrogen by Mass:

   Now, we use the formula for percentage composition:

   $$ \text{Percentage H by mass} = \left( \frac{\text{Total Mass of H}}{\text{Molar Mass of } C_2H_6} \right) \times 100\% $$

   $$ \text{Percentage H by mass} = \left( \frac{6.048 \text{ g/mol}}{30.068 \text{ g/mol}} \right) \times 100\% $$

   $$ \text{Percentage H by mass} \approx 0.2011 \times 100\% $$

   $$ \text{Percentage H by mass} \approx 20.11\% $$

Conclusion

The calculated percentage composition of hydrogen by mass in ethane ($C_2H_6$) is approximately $20.11\%$. Comparing this value to the given options, it is closest to 20%.