Question

A radioactive substance has a decay constant \(10^{-1}\) per year. What percentage of substance undergoes decay in 13.86 years?

• A. \(25\)
• B. \(50\)
• C. \(75\)
• D. \(85\)

Hint:

Find the half-life from the decay constant, then count how many half-lives fit into the given time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Radioactive decay is a first order process. The half-life is \(t_{1/2} = \dfrac{0.693}{\lambda}\). After every half-life the amount left is halved.

Step 2: Calculation of half-life
\(t_{1/2} = \dfrac{0.693}{0.1} = 6.93\) years.

Step 3: Number of half-lives
\(n = \dfrac{13.86}{6.93} = 2\).

Step 4: Amount left and decayed
Fraction left \(= \left(\dfrac{1}{2}\right)^{2} = \dfrac{1}{4}\), that is 25%.
So the decayed fraction is \(100 - 25 = 75\%\).

Final Answer:
75% of the substance decays in 13.86 years. \[ \boxed{\text{(C) }75} \]