Question

The mean lives of a radioactive substance are 1620 years and 405 years 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