Radioactive Decay; Half-Life; Decay Constant

Radioactivity is a spontaneous process. It goes on independent of external control. it is not affected by  temperature or pressure or by chemical treatment . It is also a random process as no one can predict which atom will disintegrate at a given time.

Experiments have shown that each radioactive element has a definite rate of decay which can be characterized by its Half-life.
Half-Life of a radioactive element is the time taken for half of the atoms initially present in the element to decay.
If the half-life of an element is T years, it means that after T years, 1 gm of the element will have a mass of 1/2gm, after 2T years, the mass of the element will be 1/4gm (or ½ of ½ gm) and so on.
Thus if we have 1000 atoms of a radioactive element initially, whose half-life is 10 years, then after 10 years, 500 atoms will remain ; after 20 years, 250 atoms will be left and after 30 years, 125 atoms will be left undecayed and so on.

Decay Constant, λ

The rate of decay of radioactive elements is found to be proportional to the number of atoms of the material present. Suppose there are N atoms of a radioactive element present at a time, t, then the probable number of disintegrate per unit time or activity can be expressed by –dN/dt (The minus sign arises from the fact that N is decreasing with time). Since the rate of disintegration is proportional to the number of atoms present at a given time, we have
– dN/dt α N or dN/dt = -λN
where λ is a constant of proportionality called the Decay Constant of the element.

From the above equation we have
λ = -1/N(dN/dt)
Hence Decay Constant is defined as the instantaneous rate of decay per unit atom of a substance OR
no. of atoms disintegrating per second/no. of atoms in the source at the time = λ
Also by interpreting the first equation, we have that
N = N0e-λt
where N0 is the number of atom present at time t = 0 (i.e. at the time when observations of decay were begun) and N is the number of atoms present at time t.
We obtain the time required for half of the atoms to disintegrate (half-life) by substituting N = 1/2N0 into this equation N = N0e-λt and eliminating N0 we have N0/2 = N0e-λt
½ = e –λt
Taking the natural or Naperian logarithm of both
loge ½ = – λt
But loge ½ = loge 1 – loge 2 = 0 – loge 2 = -0.693
Hence, -0.693 = -λt
t = 0.693/λ

Questions

1. Radioactivity refers to the particles which are emitted from nuclei as a result of
   A. nuclear stability  B. nuclear instability  C. Uranium decrement  D. Proton Increment
   2. Which of these is not part of radioactive emission?
   A. Alpha   B. Gamma   C. Electron  D. Beta
   3. Complete this statement, The rate of decay of radioactive elements is found to be ………………………….. to the number of atoms of the material present.
   A. Proportional  B. Inverse  C. indirect  D. direct
   4. Half-Life of a radioactive element is the time taken for half of the atoms initially present in the element to …………………………
   A. form  B. compose  C. decay  D. stimulate
   5. A certain radioactive element has a half-life of 10 years. How long will it take to loose 7/8 of its atoms originally present.
   A. 20  B. 30  C. 25  D. 35