Question

A medical lab has 16gram of sample of radioactive isotope after 6hours it found that 12gm of sample have decayed the half life of isotope is?

Answer 1

`"3 hour"`

Explanation:

General equation for radioactivity is

`N = N_0 e^(-λt) color(white)(...)....[1]`

Where

• `N_0 =` Initial amount of radioactive isotope
• `N =` Amount of radioactive isotope after time `t`
• `λ =` Radioactive decay constant

Amount of radioactive isotope left after 6 hours is, 16 g - 12 g = 4 g

`"4 g" = "16 g" × e^(-λ * "6 hr")`

`(4 cancel"g")/(16 cancel"g") = e^(-λ * "6 hr")`

`1/4 = e^(-λ * "6 hr")`

`4 = e^(λ * "6 hr")`

Apply natural log on both the sides

`ln4= "6 hr" × λ`

`λ = ln4 / "6 hr"`

`ln2/"T"_"1/2" = ln4/"6 hr" color(white)(..)[∵ λ = ln2/"T"_"1/2"]`

`cancel(ln2)/"T"_"1/2" = (2 cancel(ln2))/"6 hr" color(white)(..)[∵ lnx^n = nlnx]`

`"T"_(1//2) = "6 hr" / 2 = color(blue)"3 hr"`

ALTERNATIVE APPROACH

Equation `[1]` can also be written as

`N = N_0/(2^"n") color(white)(...)....[2]`

Where

• `"n ="` Number of half lives it takes to decay from `N_0` to `N`

`"4 g" = "16 g"/2^"n"`

`2^"n" = "16 g"/"4 g"`

`2^"n" = 4`

`2^"n" = 2^2`

`"n" = 2`

Since we got `"n = 2"`, we can say that 6 hours is 2 half lives.

∴ Half life `= "6 hr"/2 = color(blue)"3 hr"`