Question

If the mass of a radioactive sample is doubled, do (a) the activity of the sample and (b) the disintegration constant of the sample increase, decrease, or remain the same?

Short answer

1. The activity of the sample will increase.
2. The disintegration constant of the sample remains the same.

Step-by-step solution

The given data

The mass of the radionuclide sample is doubled.

Understanding the concept of decay  

The activity of a radionuclide depends on the number of nuclei and the disintegration constant of the sample. Now, according to stochiometric calculations, the concentration of a sample depends on its activity which further depends on the mass, while the disintegration constant is only dependent on the temperature factor of the decay process.

The number of atoms in a given mass of an atom,

$N=\frac{m}{A}{N}_A$ ……(i)

Here,$N_A=6.022\times {10}^{23}\frac{\mathrm{atoms}}{\mathrm{mol}}$

The concentration of the radioactive material as follows:

$\left[A_t\right]\alpha {\left[A_0\right]}^{n}or\left[A_t\right]=k{\left[A_0\right]}^n$ …… (ii)

Here,$\left[A_t\right]$is the concentration of the sample at time t,$\left[A_0\right]$is the initial concentration of the sample, n is the order of the decay and is the disintegration constant of the sample.

a) Calculate the activity of the sample

From equation (i), it can be clearly seen that the number of undecayed nuclei in the sample doubles with mass being doubled $\left(N\alpha m\right)$.

Now, this implies that the concentration of the sample doubles and hence using equation (ii), we can get that the activity of the sample also increases. That is given as:

$\left[A_t\right]\alpha {\left[2A_0\right]}^n$

Thus, the activity of the sample increases.

b) Calculate the disintegration constant

From equation (ii), it is clear that the disintegration is inversely proportional to the activity change or the concentration change of the sample. But as it only depends on the temperature factor, it remains the same irrespective of any mass or concentration change.

Hence, the disintegration constant of the sample remains the same.