GATE 2022 BT Q.16: solution and answer

GATE 2022 · BTQ.16 · 1 mark · MCQ

The overall stoichiometry for an aerobic cell growth is

3 C_{6} H_{12} O_{6} + 2.5 NH_{3} + O_{2} \to 1.5 C_{a} H_{b} O_{c} N_{d} + 3 CO_{2} + 5 H_{2} O

What is the elemental composition formula of the biomass?

(A)
$C_{9} H_{18.2} O_{5} N_{1.667}$
(B)
$C_{9} H_{22.33} O_{6} N_{1.667}$
(C)
$C_{10} H_{18.2} O_{5} N_{1.667}$
(D)
$C_{10} H_{22.33} O_{6} N_{1.667}$

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Answer and solution

ANSWER

The biomass formula is $C_{10} H_{22.33} O_{6} N_{1.667}$ , option (D).

Official answer: DSource: Official answer key, GATE 2022 BT (IIT Kharagpur)

Element balance: each of C, H, O and N must have the same number of atoms on both sides. The biomass term is $1.5 C_{a} H_{b} O_{c} N_{d}$ , so for each element, the atoms in the biomass equal (atoms in the reactants $-$ atoms in the other products), and dividing by $1.5$ gives the subscript.
Carbon: the reactants carry $3 \times 6 = 18$ ; $3 CO_{2}$ carries $3$ . So $1.5 a = 18 - 3 = 15$ , which gives $a = 10$ .
Hydrogen: the reactants carry $3 \times 12 + 2.5 \times 3 = 36 + 7.5 = 43.5$ ; $5 H_{2} O$ carries $5 \times 2 = 10$ . So $1.5 b = 43.5 - 10 = 33.5$ , which gives $b = 22.33$ .
Oxygen: the reactants carry $3 \times 6 + 1 \times 2 = 20$ (one $O_{2}$ molecule is 2 atoms); $3 CO_{2}$ carries $6$ and $5 H_{2} O$ carries $5$ . So $1.5 c = 20 - 11 = 9$ , which gives $c = 6$ .
Nitrogen: all of it comes from $2.5 NH_{3}$ , and no other product contains N. So $1.5 d = 2.5$ , which gives $d = 1.667$ .
Biomass formula: $C_{10} H_{22.33} O_{6} N_{1.667}$ , which is option (D). As a check, only C and D have $a = 10$ , and of those only D has $b = 22.33$ and $c = 6$ .

KEY IDEA

Atoms are conserved, so balance each element (C, H, O, N) separately. Whatever the reactants supply and the $CO_{2}$ and $H_{2} O$ do not take away ends up in the biomass. Divide that remainder by the biomass coefficient, $1.5$ , to get each subscript.

WATCH OUT FOR

The common slips are counting $O_{2}$ as one oxygen atom instead of two (that gives $c = 5.33$ ), forgetting the hydrogen that $NH_{3}$ brings in (that gives $b = 17.33$ ), and forgetting to divide by the biomass coefficient $1.5$ (that gives $C_{15}$ ). None of these results is among the options, so each one sends you to a wrong guess.

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GATE 2022 BT Q.16: solution and answer

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