GATE 2025 CY – Question 58

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

An archaeological $^{14}$C specimen gives 45 counts per gram of carbon in 5 minutes. Fresh wood gives 20 counts per gram per minute. Background is 5 counts per minute. The specimen age in years (integer) is _____. Given $t_{1/2}=5730$ years.

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Correct answer: 10926

Explanation

**Radiocarbon dating** relies on the decay of $^{14}$C: the activity falls by half every half-life, $A=A_0\,2^{-t/t_{1/2}}$.

**Step 1: net activities (counts per gram per minute).**
- Specimen: 45 counts in 5 min per gram, so $9$ counts/min; subtract the background of 5: $9-5=4$.
- Fresh wood: $20-5=15$ counts/min.

**Step 2: age.**
$$\frac{A}{A_0}=\frac4{15}=2^{-t/5730}\;\Rightarrow\;t=5730\,\log_2\frac{15}{4}.$$

$$\log_2(3.75)=1.9069,\qquad t=5730\times1.9069=\mathbf{10\,926\ years}.$$