GATE 2023 BT Q.46: solution and answer

Official answer: 0.41 to 0.44Source: Official answer key, GATE 2023 BT (IIT Kanpur)

1. Definition of respiratory quotient: $$\text{RQ}=\frac{{\text{moles of CO}}_2\text{ evolved}}{{\text{moles of O}}_2\text{ consumed}}$$ Both are taken per mole of substrate, so RQ is a ratio in ${\text{mol mol}}^{-1}$ and has no units. Here the substrate is hexadecane, $C_{16}{H}_{34}$.

2. Read the two coefficients from the stoichiometric equation. Per mole of $C_{16}{H}_{34}$: ${\text{O}}_2$ consumed $=12.5\text{mol}$ and ${\text{CO}}_2$ evolved $=5.37\text{mol}$.

3. Check the carbon balance to confirm the coefficients are consistent: $16=10.6(\text{biomass, 1 C each})+5.37({\text{CO}}_2)=15.97\approx 16$.

4. Compute: $$\text{RQ}=\frac{5.37}{12.5}=0.4296\approx 0.43$$

5. Why it is so low: hexadecane is a highly reduced substrate. It has no oxygen and $\text{H/C}=34/16=2.125$. A large share of the ${\text{O}}_2$ goes into oxidising hydrogen to water ($11.4{\text{mol H}}_2\text{O}$), and about two-thirds of the carbon ($10.6/16\approx 0.66$) ends up in biomass instead of ${\text{CO}}_2$. Both effects push RQ well below 1, which is typical of growth on alkanes.