Question

a sample start with 1000 radioactive atoms .how many half lives have elapsed when 750 atoms have decayed?​

Answer 1

Answer:

250

Explanation:

1000-750=250

half 250

Answer 2

Answer:

A sample starts with 1000 radioactive atoms and  750 atoms have decayed so the number of half-lives is equal to $0.38$

Explanation:

• The half-life of a sample is the time required to react half of the sample, it is given by the formula

                   $T_{1/2}=\frac{0.693}{\lambda }$    λ-decay constant (equation 1)

• Consider a sample with a half-life $T_{1/2}$ and  $N(0)$ be the beginning amount of sample at a later time $t$ the amount of sample would be

                 $N\left(t\right)=N\left(0\right)\ e^{-\lambda t}$ (equation 2)

From the question, we have

the starting amount of the sample $N(0)=1000 atoms$

the amount that remained after a time $t$ is $N(t)=750atoms$

rearrange equation 2 with equation 1

                 $N\left(t\right)=N\left(0\right)\ e^{-\0.693* t/T_{1/2} }$

⇒              $N\left(t\right)/N(0)= e^{-\0.693* t/T_{1/2} }$

                $N\left(0\right)/N(t)= e^{\0.693* t/T_{1/2} }$ (equation 3)

by taking logarithm on both sides of equation 3

              $ln(N\left(0\right)/N(t))= ln(e^{\0.693* t/T_{1/2} })$

             $ln(N\left(0\right)/N(t))= {\0.693* t/T_{1/2} }$

since exponential and logarithm are inverse functions they cancel each other

now substitute the given values

             $ln(1000/ 750)= {\0.693* t/T_{1/2} }\\0.26/0.693=t/T_{1/2}$

the number of half-lives $t/T_{1/2}=0.26/0.693=0.38$