Question

A certain element has a half life of

2.52.5

billion years.

a. You find a rock containing a mixture of the element and lead. You determine that

55?%

of the original element? remains; the other

9595?%

decayed into lead. How old is the?rock?

b. Analysis of another rock shows that it contains

6060?%

of its original? element; the other

4040?%

decayed into lead. How old is the? rock?

Answer 1

The formula for computing the decay in a radioactive substance is N_t = N₀ e^-kt or, N_t/N₀ = e^-kt,where N₀ is the initial quantity,N_t is the quantity after t years and k is the constant of decay. Here t is measured in billions of years. Also, the substance has a half-life of 2.5 billion years so that ½ = e^-k*2.5 . On taking natural log of both the sides, we get -2.5k ln e = ln0.5 or, -2.5 k = -0.69314718 so that k =0.69314718/2.5 = 0.277258872.

A. Let the rock be t billion years old. Then 0.05 = e^{-0.277258872t}. On taking natural log of both the sides, we get -0.277258872 t = ln 0.05 = -2.995732274 so that t = 2.995732274/0.277258872 = 10.80482025 = 10.80 ( on rounding off to 2 decimal places). Thus, the rock is 10.80 billion years old.

B. Let the rock be t billion years old. Then 0.60 = e^{-0.277258872t}. On taking natural log of both the sides, we get -0.277258872 t = ln 0.60 =-0.510825623 so that t =0.510825623/0.277258872 = 1.842413987 = 1.84 ( on rounding off to 2 decimal places). Thus, the rock is 1.84 billion years old.