The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 80% of the lead to decay

Therefore it will take 7.66 hours for 80% of the lead decay.

Explanation:

The differential equation for decay is

Integrating both sides

ln A= kt+c₁

        []

The initial condition is A(0)= A₀,

.........(1)

Given that the  has half life of 3.3 hours.

For half life putting this in equation (1)

    [taking ln both sides, ]

⇒k= - 0.21

Now A₀= 1 gram, 80%=0.8

and A= (1-0.8)A₀ = (0.2×1) gram = 0.2 gram

Now putting the value of k,A and A₀in the equation (1)

⇒ t≈7.66

Therefore it will take 7.66 hours for 80% of the lead decay.

explanation: