GATE 2023 CE (CE2) – Question 64
The theoretical aerobic oxidation of biomass $(C_5H_7O_2N)$ is given below: $C_5H_7O_2N+5O_2\rightarrow5CO_2+NH_3+2H_2O$. The biochemical oxidation of biomass is assumed as a first-order reaction with a rate constant of 0.23/d at 20 °C (logarithm to base e). Neglecting the second-stage oxygen demand from its biochemical oxidation, the ratio of BOD₅ at 20 °C to total organic carbon (TOC) of biomass is ______ (round off to two decimal places). [Consider the atomic weights of C, H, O and N as 12 g/mol, 1 g/mol, 16 g/mol and 14 g/mol, respectively]
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Correct answer: 1.81 to 1.83
Explanation
**Step 1: molar masses.** Biomass $C_5H_7O_2N$: $5(12)+7(1)+2(16)+14=113$ g/mol. Organic carbon in one mole: $5\times12=60$ g.
**Step 2: ultimate oxygen demand.** From $C_5H_7O_2N+5O_2\to5CO_2+NH_3+2H_2O$, one mole needs 5 mol of $O_2$:
$$BOD_u=5\times32=160\text{ g per mole of biomass}.$$
**Step 3: BOD exerted in 5 days.** For first-order decay,
$$\frac{BOD_5}{BOD_u}=1-e^{-kt}=1-e^{-0.23\times5}=1-e^{-1.15}=1-0.3166=0.6834 .$$
$$BOD_5=160\times0.6834=109.3\text{ g}.$$
**Step 4: ratio to TOC.**
$$\frac{BOD_5}{TOC}=\frac{109.3}{60}=\mathbf{1.82}.$$