The Radioactive Decay law states that the number of nuclei undergoing decay per unit time is directly proportional to the number of unchanged nuclei present at that instant.

The formula for Radioactive Decay law is given as,

– (dN / dt) ∝ N

– (dN / dt) = λ N

So we can say that at a time t = 0 we have N₀ radioactive nuclei present, substituting the current values in the above formula we get the number of radioactive nuclei present at time t.

So the formula for number of nuclei present at any instant is N = N₀ e^-λt where
N = number of atoms
λ = decay constant
N0 = number of nuclei when time is 0 (the initial quantity)
λ = decay constant
t = time

The Radioactive Decay law states that the number of nuclei undergoing decay per unit time is directly proportional to the number of unchanged nuclei present at that instant.

The formula for Radioactive Decay law is given as,

– (dN / dt) ∝ N

– (dN / dt) = λ N

dN = the number of decay events

dt = interval of time

N = number of atoms

λ = decay constant

Therefore, in a given sample of a particular radioisotope, the number of decay events (dN) expected to occur in a small interval of time (dt) is proportional to the number of atoms present (N). The negative sign indicates that the number of atoms of radioactive element decreases with time.

An unstable nucleus contains excess of neutrons, and this causes neutron transforming into proton by emitting an electron.

When a β particle is emitted by an atom, its atomic number increases by 1 and mass number doesn’t change.

Rutherford and Soddy’s law of β decay states that “whenever an β particle is emitted from a radioactive nucleus, the atomic number decreases by 1 and the mass number remains the same and the position of the element shifts by one place to right in the periodic table”.

When an α (alpha) particle is emitted by an atom, its atomic number (Z) decreases by 2 and mass number (A) decreases by 4.

Here P is called the parent nucleus.
D is called as daughter nucleus.

In the above example Uranium is the parent nucleus and thorium is the daughter nucleus.

Rutherford and Soddy’s law of α emission states that “whenever an α particle is emitted from a radioactive nucleus, the atomic number decreases by 2 and the mass number decreases by 4 and the position of the element shifts by two places to left in the periodic table”.

Mass defect is the difference between the actual mass of nucleus and sum of masses of constituent nucleons.

The formula for Mass defect is given as,
m = [Z m_p + (A – Z)m_n] – M

M = mass of nucleus
A = mas number                           
m_p = mass of proton
Z = atomic number                              
(A – Z) = number of neutron
m_n = mass of neutron