Derive an expression that relates the half-life and the decay constant for a sample of aparticular radioactive nucleus. Calculate the half-life and decay constant for a radioactiveisotope sample if the initial number of radioactive atoms is 450 and this reduces to 63 after24.3 hours.

The relationship between half-life (T) and decay constant (λ) for a radioactive substance can be derived from the equation of radioactive decay, which is:

N = N0 * e^(-λt)

Where: N is the number of atoms at time t, N0 is the initial number of atoms, λ is the decay constant, t is the time.

The half-life is the time it takes for half of the radioactive atoms to decay. So, when t = T, N = N0/2. Substituting these into the decay equation gives:

N0/2 = N0 * e^(-λT)

Dividing both sides by N0 gives:

1/2 = e^(-λT)

Taking the natural logarithm of both sides gives:

ln(1/2) = -λT

So:

λ = -ln(1/2) / T

This is the relationship between the decay constant and the half-life.

Now, let's calculate the half-life and decay constant for the given radioactive isotope sample. The initial number of radioactive atoms is 450 and this reduces to 63 after 24.3 hours. So, N0 = 450, N = 63, and t = 24.3 hours.

Substituting these into the decay equation gives:

63 = 450 * e^(-λ * 24.3)

Dividing both sides by 450 gives:

63/450 = e^(-λ * 24.3)

Taking the natural logarithm of both sides gives:

ln(63/450) = -λ * 24.3

So:

λ = -ln(63/450) / 24.3

Once you have λ, you can substitute it into the relationship derived earlier to find T:

T = -ln(1/2) / λ

These are the steps to derive the relationship between half-life and decay constant, and to calculate these quantities for a specific radioactive isotope sample.