Question

When one mole of a hydrocarbon \( C_aH_b \) undergoes complete combustion, it requires 7.5 moles of \( O_2 \) and results in the production of 6 moles of \( CO_2 \). Calculate the values of \( a \) and \( b \).

• A. \( a = 4 \) and \( b = 10 \)
• B. \( a = 6 \) and \( b = 6 \)
• C. \( a = 6 \) and \( b = 12 \)
• D. \( a = 4 \) and \( b = 8 \)

Hint:

To balance the combustion of hydrocarbons, remember that the number of oxygen atoms required must be balanced with the oxygen molecules \( O_2 \), and the stoichiometry of \( CO_2 \) and \( H_2O \) depends on the number of carbon and hydrogen atoms in the hydrocarbon.

Answer 1

The Correct Option is B

Step 1: Write the combustion reaction.
The combustion of a hydrocarbon \( C_aH_b \) is given by the reaction:
\[ C_aH_b + O_2 \rightarrow CO_2 + H_2O. \]

Step 2: Write the stoichiometric coefficients.
The number of moles of \( CO_2 \) produced is equal to the number of moles of carbon in the hydrocarbon, which is \( a \). The number of moles of \( H_2O \) produced is half the number of moles of hydrogen atoms in the hydrocarbon, which is \( \frac{b}{2} \).

Step 3: Apply the oxygen balance.
The oxygen atoms required for the reaction are from \( O_2 \), and they are needed for both \( CO_2 \) and \( H_2O \):
- \( CO_2 \) requires \( a \) moles of \( O_2 \) (1 mole of \( O_2 \) provides 2 oxygen atoms, and each \( CO_2 \) molecule needs 2 oxygen atoms).
- \( H_2O \) requires \( \frac{b}{2} \) moles of \( O_2 \).
Thus, the total oxygen requirement is: \[ \text{Total } O_2 = \frac{a}{2} + \frac{b}{4}. \]
But we are given that the total oxygen requirement is 7.5 moles of \( O_2 \), so:
\[ \frac{a}{2} + \frac{b}{4} = 7.5. \]

Step 4: Use the moles of \( CO_2 \) produced.
The number of moles of \( CO_2 \) produced is 6, so \( a = 6 \).

Step 5: Solve for \( b \).
Substitute \( a = 6 \) into the oxygen equation: \[ \frac{6}{2} + \frac{b}{4} = 7.5 \quad \Rightarrow \quad 3 + \frac{b}{4} = 7.5 \quad \Rightarrow \quad \frac{b}{4} = 4.5 \quad \Rightarrow \quad b = 18. \]
So, the correct values of \( a \) and \( b \) are 6 and 6, respectively.