Question

A radioactive substance which emits alpha particle follows a first-order reaction. The half-life period of this radioactive substance is 30 hours. Calculate the fraction in percent (%) of the radioactive substance which remains after 90 hours.

Answer 1

Given,

Half life `(t_(1/2))` = 30 hr

To calculate the fraction (in %) after 90 hr

For first-order kinetics

`t_(1/2) = 0.693/"K"`

`K = 0.693/(30 "hr")`

= 0.0231 hr⁻¹

`t = 2.303/K log  ([A_0])/([A_t])`

`90 = 2.303/0.0231 log  ([100])/([A_t])`

`log  ([100])/([A_t]) = (90 xx 0.0231)/2.303`

= 0.9027

log[100] − log[A_t] = 0.9027

log[A_t] = log[100] − 0.9027

= 2 − 0.9027

= 1.0973

[A_t] = antilog₁₀ (1.0973)

= 12.5

Fraction = 100 − 12.5

= 87.5%

After 90 hours, 87.5% of the radioactive substance will still be present.