Radioactive elements decay via first-order kinetics. Consider a certain type of nucleus that has a rate constant of 1.9× 10^-2 h^-1. A sample contains 6.6× 10⁸ radioactive nuclides. Calculate the time required to reduce that number to 2.8× 10⁸. A) 4.5 h B) 2.0 × 10¹⁰ h C) 4.5 × 10¹ h D) 2.0 × 10¹ h E) 8.1 × 10^-1 h

Question

Quizplus · Accepted Answer

The decay process of the radioactive element follows first-order kinetics, which can be described by the following equation:

ln(Nt/N0) = -kt

where Nt is the number of radioactive nuclides at time t, N0 is the initial number of radioactive nuclides, k is the rate constant, and t is time.

We are given N0 = 6.6×10^18 and Nt = 2.8×10^18. We need to find the time required to reduce the number of radioactive nuclides from N0 to Nt. We can rearrange the above equation as:

t = (ln(N0/Nt))/k

Substituting the given values, we get:

t = (ln(6.6×10^18/2.8×10^18))/(1.9×10^-12 h^-1)
t = 4.5×10^10 seconds
t = 4.5×10^10/3600 hours
t = 1.25×10^7 hours
t = 4.5×10^1 hours

Therefore, the time required to reduce the number of radioactive nuclides from 6.6×10^18 to 2.8×10^18 is 4.5×10^1 hours or 45 hours. The best choice is C, 4.5×10^11 hours.

Radioactive Elements Decay Via First-Order Kinetics