Question

The half life of radium is 1600 years. The fraction of a sample of radium that would remain after 6400 years:
A. $\dfrac{1}{4}$
B. $\dfrac{1}{2}$
C. $\dfrac{1}{8}$
D. $\dfrac{1}{16}$

Answer 1

To solve the given question, we must know the formula for the number of atoms left in a radioactive element at a given time, when the radioactive element is decaying. The half life of a radioactive element is the time required for decaying of the element to half of the initial number of atoms.

Complete step by step answer:
Radium is a radioactive element. A radioactive element is a very unstable element. Since the atoms of a radioactive element are unstable, they lose their energy and decay. During a decay of a radioactive element, the number of atoms left in the element at given time t is given as $N={{N}_{0}}{{e}^{-\lambda t}}$ ….. (i),
where N is the number of atoms left in the element at that time t, ${{N}_{0}}$ is the number of atoms that were present in the element initially (i.e. at time $t=0$) and $\lambda$ is the decay constant for the element.

Let us now understand what meant by half the life of a radioactive element. The half life (${{t}_{1/2}}$) of a radioactive element is the time required for decaying of the element to half of the initial number of atoms. In other words, it is the value of t for which $N=\dfrac{{{N}_{0}}}{2}$.
Now, substitute $N=\dfrac{{{N}_{0}}}{2}$ and $t={{t}_{1/2}}$ in (i).
$\Rightarrow \dfrac{{{N}_{0}}}{2}={{N}_{0}}{{e}^{-\lambda {{t}_{1/2}}}}$
$\Rightarrow \dfrac{1}{2}={{e}^{-\lambda {{t}_{1/2}}}}$
$\Rightarrow 2={{e}^{\lambda {{t}_{1/2}}}}$

Now, take ln on both sides.
$\Rightarrow \ln 2=\ln \left( {{e}^{\lambda {{t}_{1/2}}}} \right)$
$\Rightarrow \ln 2=\lambda {{t}_{1/2}}$
$\Rightarrow {{t}_{1/2}}=\dfrac{\ln 2}{\lambda }$
Hence, we found an expression of the half life.
But, it is given that ${{t}_{1/2}}=1600years$
$\Rightarrow 1600=\dfrac{\ln 2}{\lambda }$
$\Rightarrow \lambda =\dfrac{\ln 2}{1600}\text{per year}$
Now, to find the fraction of the sample that remains after 6400 years substitute the value of $\lambda$ and $t=6400years$.
$\Rightarrow N={{N}_{0}}{{e}^{-\dfrac{\ln 2}{1600}(6400)}}$
$\Rightarrow N={{N}_{0}}{{e}^{-4\ln 2}}$
We can write the above equation as:
$N={{N}_{0}}{{e}^{\ln {{2}^{-4}}}}$
$\Rightarrow N={{N}_{0}}\left( {{2}^{-4}} \right)$
$\therefore \dfrac{N}{{{N}_{0}}}={{2}^{-4}}$.
This means that the fraction of samples left after 6400 year is ${{2}^{-4}}$.

Hence,option D is the correct answer.

Note: For a given element, the decay constant is constant. Therefore, we do not have to derive the expression for the half life. We can directly use it if we know it. The term $-\lambda t$ is called power or exponent. And a power is just a number without a dimension. Therefore, in any formula which involves a power, the quantity that is used in the power must be dimensional. Therefore, the unit of decay constant is ${{s}^{-1}}$.