Question

A radioactive element X converts into another stable element Y. half life of X is 2hrs. Initially, only X is present. After time t, the ratio of atoms of X and Y is found to be 1:4, then t in hours:
A) 2
B) 4
C) Between 4 and 6
D) 6

Answer 1

This can be solved by using a decay law which tells us the number of atoms of the original element decreases continuously with time. Then the rate of disintegration also decreases.

Complete step-by-step answer:
Given, radioactive element x converts into stable element Y.
Half life of radioactive element X is given by ${T_{1/2}} = 2hrs$
Let ${N_y}$ be the number of atoms that decay in time t hrs. And Nx be the number of atoms left out in time t hrs.
From radioactive decay law, we have
Then ${N_x} = {N_0}{e^{ - \lambda t}}$ ……………….. (1)
${N_0}$ is the number of atoms of radioactive element present initially (t=0)
${N_y} = \left( {{N_0} - {N_x}} \right) = {N_0}\left( {1 - {e^{ - \lambda t}}} \right)$ ………………….. (2)
Now divide equation (1) by (2) we get,
$\dfrac{{{N_x}}}{{{N_y}}} = \dfrac{{{e^{ - \lambda t}}}}{{1 - {e^{ - \lambda t}}}} = \dfrac{1}{4}$
$4{e^{ - \lambda t}} = 1 - {e^{ - \lambda t}}$
Simplifying the above equation we get,
$5{e^{ - \lambda t}} = 1$
Rearrange the above equation
${e^{ - \lambda t}} = \dfrac{1}{5}$
To reduce that minus sign in exponential take reciprocal on both sides we get,
${e^{\lambda t}} = 5$
Now exponential is moving other side and it becomes natural logarithm that is $\ln 5$ or${\log _e}5$ (logarithm to the base e) then we get,
$\lambda t = \ln 5$
$t = 4.64hrs.$ ($\because {T_{1/2}} = \dfrac{{\ln 2}}{\lambda }$ )
After time 4.65hrs the ratio of atoms of X and Y is found to be 1:4.
Thus the correct option is ( C).

Additional Information:
Radioactivity was discovered by Henry Becquerel in 1896. He found that Uranium salt wrapped up in paper emitted certain penetrating radiation which affected a photographic plate. From this it observed that the nuclei of all heavy elements with Z above 82 are unstable. The unstable nuclei reduce their energy and become more stable by emitting certain radiation. This phenomenon of emitting certain radiation by heavy elements is called radioactivity.
Rutherford and Soddy discovered experimentally that an atom of a radioactive element disintegrates continuously giving rise to new elements. The number of atoms of the original element decreases continuously with time. Then the rate of disintegration also decreases.
According to radioactive decay law, the rate of disintegration of atoms at any instant of time is directly proportional to the number of radioactive atoms present at that instant.
Let, ${N_0}$ be the number of atoms present initially in the radioactive sample. After time t, the number of radioactive atoms left undecayed in the sample is N. according to decay law,
$\dfrac{{dN}}{{dt}}\alpha N$
$\therefore \dfrac{{dN}}{{dt}} = - \lambda N$
Here $\lambda$ is a constant of proportionality and is called decay constant or disintegration constant. The negative sign shows that N is decreasing with time.

Note: Here, the main thing is the number of atoms decayed and undecayed is very important to list out.
The process of spontaneous disintegration of the nuclei of elements with the emission of certain types of radiation is called Radioactivity.