GATE 2026 CY – Question 46

Physical Chemistry · Kinetics and Reaction Dynamics: Elementary, parallel, opposing, and consecutive reactions. Steady state approximation. Mechanism of complex reactions. Unimolecular reactions. Potential energy surface and classical trajectories, Concept of saddle points, Transition state theory: Eyring equation, thermodynamic aspects. Kinetics of polymerization. Catalysis concepts and enzyme catalysis. Kinetic isotope effects. Fast reaction kinetics: relaxation and flow methods. Diffusion controlled reactions. Kinetics of unimolecular and bimolecular photophysical processes, Quantum yield calculation, static and dynamic quenching. · 2 marks · Multiple choice

For a reaction between neutral molecules X and Y in a solution at temperature T, the measured rate of reaction is equal to the rate of diffusion. Assume X is stationary and Y is moving. If the diameter of the molecule X is five times that of Y, then the rate constant for the reaction is (η is viscosity of the solvent; k is Boltzmann constant)

$8kT/(3\eta)$
$4kT/\eta$
$24kT/(5\eta)$
$15kT/(4\eta)$

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Correct answer: (B) $4kT/\eta$

Explanation

**Diffusion-controlled rate constant (Smoluchowski).** The rate constant for the encounter of two species is
$$k_d=4\pi\,(r_X+r_Y)\,(D_X+D_Y)\,N_A\quad(\text{per mole}).$$

X is stationary ($D_X=0$), so only Y diffuses:
$$k_d=4\pi\,(r_X+r_Y)\,D_Y .$$

**Stokes-Einstein** gives the diffusion coefficient of Y:
$$D_Y=\frac{kT}{6\pi\eta r_Y} .$$

**Sizes.** The diameter of X is five times that of Y, so $r_X=5r_Y$ and $r_X+r_Y=6r_Y$:
$$k_d=4\pi\,(6r_Y)\,\frac{kT}{6\pi\eta r_Y}=\frac{4kT}{\eta}\quad(\text{option B}).$$