GATE 2022 BT Q.48: solution and answer

GATE 2022 · BTQ.48 · 2 marks · MSQ

Which of the following statements are CORRECT for an enzyme entrapped in a spherical particle?

(A)
Effectiveness factor is ratio of the reaction rate with diffusion-limitation to the reaction rate without diffusion-limitation
(B)
Internal diffusion is rate-limiting at low values of Thiele modulus
(C)
Effectiveness factor increases with decrease in Thiele modulus
(D)
Internal diffusion-limitation can be reduced by decreasing the size of the particle

See the question in the paper

Answer and solution

ANSWER

Options A, C and D are correct. The effectiveness factor is the rate with diffusion limitation divided by the rate without it. It rises toward 1 as the Thiele modulus falls, and because the modulus is proportional to particle radius, a smaller particle reduces the internal diffusion limitation. B is wrong: diffusion limits the rate at HIGH Thiele modulus, not low.

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

Definition of the internal effectiveness factor: $η = \frac{observed rate (with internal diffusion limitation)}{rate if the whole particle saw the surface concentration S _{s} (no diffusion limitation)}$ , so $0 < η \leq 1$ . Statement (A) is exactly this definition, so (A) is correct.
Thiele modulus for a sphere of radius $R$ (first-order limit of Michaelis-Menten kinetics, rate constant $k_{1} = V_{ma x} / K_{m}$ , where $V_{ma x}$ is the maximum reaction rate per unit particle volume, $K_{m}$ the Michaelis constant and $D_{e}$ the effective diffusivity of substrate inside the particle): $ϕ = \frac{R}{3} \frac{V _{ma x}}{K _{m} D _{e}}$ It compares the intrinsic reaction rate with the rate of internal diffusion; a large $ϕ$ means reaction is fast compared with diffusion.
For a sphere with first-order kinetics: $η = \frac{1}{ϕ} (\frac{1}{tanh 3 ϕ} - \frac{1}{3 ϕ})$ Limits: $ϕ \to 0 \Rightarrow η \to 1$ (reaction controls, substrate reaches the whole particle); $ϕ ≫ 1 \Rightarrow η \approx 1/ ϕ$ (diffusion controls, only an outer shell of the particle works).
So internal diffusion is rate-limiting at HIGH values of $ϕ$ , not low ones. Statement (B) is incorrect.
$η$ falls steadily as $ϕ$ rises, so a decrease in $ϕ$ raises $η$ toward 1. Statement (C) is correct.
Since $ϕ \propto R$ , a smaller particle gives a smaller $ϕ$ and hence a larger $η$ : the internal diffusion limitation is reduced (the substrate has a shorter path to the enzyme at the centre). The same direction holds for general Michaelis-Menten kinetics, whose modulus is also proportional to $R$ . Statement (D) is correct.
Correct options: A, C, D.

KEY IDEA

The Thiele modulus $ϕ = \frac{R}{3} V_{ma x} / (K_{m} D_{e})$ measures reaction speed against internal diffusion speed. The effectiveness factor $η$ (rate with diffusion limitation divided by rate without it) equals 1 when $ϕ$ is small and falls roughly as $1/ ϕ$ when $ϕ$ is large. Since $ϕ \propto R$ , making the particle smaller raises $η$ .

WATCH OUT FOR

Getting the direction of the Thiele modulus backwards and marking B: a LOW $ϕ$ means diffusion is fast compared with reaction ( $η \approx 1$ , reaction controls). A second slip is reading the effectiveness factor the other way up (rate without limitation over rate with it), which would give a value above 1.

From the GATE 2022 BT paper. Every question of this paper · Download the full paper · All 2 Immobilisation questions, with notes