Question

The mean lives of a radioactive substance are 1620 year and 405 year for a-emission and b-emission respectively. Find the time during which three-fourth of a sample will decay if it is decaying both by a-emission and b-emission simultaneously.

• A. 249 years
• B. 449 years
• C. 133 years
• D. 99 years

Answer 1

Answer: (B)

449 years

Answer 2

The decay constant l is the reciprocal of the mean life t. Thus, l_a = $\frac { 1 } { 1620 }$ per year and

l_b = $\frac { 1 } { 405 }$per year \Total decay constant, l = l_a + l_b or

l = $\frac { 1 } { 1620 } + \frac { 1 } { 405 } = \frac { 1 } { 324 }$ per year

We know that N = N₀ e^–lt When $\frac { 3 } { 4 }$ th part of the sample has disintegrated, N = N₀/4

\ = N₀e^–lt or e^lt = 4 Taking logarithm of both sides, we get lt = log_e 4 or t = $\frac { 1 } { \lambda }$log_e 2² ₌ $\frac { 2 } { \lambda }$ log_e 2

= 2 × 324 × 0.693 = 449 year