de-Broglie’s hypothesis states value of a wavelength associated with a particle of mass m which is moving with speed v.

The formula for de-Broglie’s hypothesis is given as,

Wavelength = Planck’s constant / momentum

λ = h / p…………………..(1)

We know that momentum = mass × velocity

substituting value of p in (1) we get,

 λ = h / mv

 λ = wavelength                                     h = Planck’s constant

p = momentum                                     m = mass

v = velocity

Half-life period is used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive a radioactive decay. It is the time required for a quantity to reduce to half its initial value.

we know N = N₀ e^-λt and half life T = t  , N = N₀ / 2

substituting these values we get,

N₀/ 2 = N₀ e^-λt

1/ 2 = e^-λt

e^λt = 2

λ T = log _e 2

λ T = 0.693 —– value of log _e 2 is 0.693

T = 0.693 /λ

Where

T = half-life period

λ =the decay constant of the decaying quantity.

The decay constant is defined as the ratio of amount of substance decayed per unit time to amount of substance present at that time.

we know ,

N = N₀ e^-λt

and t = 1 /λ

Substitute t= 1 /λ in above formula we get

N = N₀ e^-λ(1/λ)

N = N₀ e^-1

N = N₀ / e

N = N₀ /2.718

N = 0.37 N₀

Now decay constant can also be defined as the reciprocal of the duration in which a substance decays at 37% of its original quantity.

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.