GATE 2025 CY – Question 65

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 · Numerical answer

The Lineweaver-Burk plot for an enzyme obeying the Michaelis-Menten mechanism is given below. The slope of the line is $0.36\times10^{-2}$ s, and the y-intercept is $1.20\ \mathrm{mol^{-1}L\,s}$. The value of the Michaelis constant ($K_M$) is ____ $\times10^{-3}\ \mathrm{mol\,L^{-1}}$ (in integer). [Note: $v$ is the initial rate, and $[S]_0$ is the substrate concentration]

Lineweaver–Burk plot of reciprocal velocity against reciprocal substrate concentration.

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Correct answer: 3

Explanation

The **Lineweaver-Burk** (double-reciprocal) equation of the Michaelis-Menten mechanism is
$$\frac1v=\frac{K_M}{V_{max}}\cdot\frac1{[S]_0}+\frac1{V_{max}} .$$

This is a straight line of $1/v$ against $1/[S]_0$:
- the **slope** is $\dfrac{K_M}{V_{max}}$,
- the **y-intercept** is $\dfrac1{V_{max}}$.

**Divide the slope by the intercept** to eliminate $V_{max}$:
$$K_M=\frac{\text{slope}}{\text{intercept}}=\frac{0.36\times10^{-2}\ \text{s}}{1.20\ \text{mol}^{-1}\text{L s}}=3.0\times10^{-3}\ \text{mol L}^{-1}.$$

$K_M=\mathbf{3}\times10^{-3}\ \text{mol L}^{-1}$.