Question

A radioactive substance is known to decay at a rate proportional to the amount present. If initially there is 100
milligrams of the radioactive substance present and after two hours it is observed that the radioactive substance has lost 10 percent of its original mass, find the half life of the radioactive substance.

Answer 1

Step-by-step explanation:

let the amount of radioactive material be α

Given that,⟹−

dt

dα

=kα

Where k is the proportionality constant

⟹

α

1

dα=

k

−1

dt

let α

0

be the initial amount taken,

solving the differential equation gives,

α=α

0

e

−kt

Given that after an hour 10percent of the given material has been decayed,

substituting this in the obtained equation gives,

10

9α

0

=α

0

e

−k

Thus, we can obtain the value of k from this equation.

k=0.105379

Now for finding the half life,

2

α

0

=α

0

e

−tk

∴t

2

1

=

k

0.693

⟹t

2

1

=

0.105379

0.693

∴t

2

1

=6.58hrs