Question

Given specimens uranium\({\rm{ - 232}}\)(\({{\rm{t}}_{{\rm{1/2}}}}{\rm{ = 68}}{\rm{.9 y}}\)) and uranium\({\rm{ - 233}}\)(\({{\rm{t}}_{{\rm{1/2}}}}{\rm{ = 159,200 y}}\)) of equal mass, which one would have greater activity and why?

Short answer

The activity is inversely proportional to the half-life, which indicates that the higher the half-life, the lower the activity. Uranium\({\rm{ - 232}}\) has a shorter half-life than uranium\({\rm{ - 233}}\), which means it has higher activity.

Step-by-step solution

Step 1: Nuclear Chemistry

Nuclear chemistry is a branch of chemistry that studies radioactivity, nuclear processes, and atomic nuclei alterations such as nuclear transmutation and nuclear characteristics.

Explanation

The next formula is used here for this task as:

\({\rm{t = }}\frac{{\rm{1}}}{{\rm{\lambda }}}{\rm{In}}\frac{{{{\rm{n}}_{\rm{0}}}}}{{{{\rm{n}}_{\rm{t}}}}}\)

As the atomic masses of both substances are the same, the nuclide concentration will be the same.

The decay constant has a formula as well:

\({\rm{\lambda = }}\frac{{{\rm{In(2)}}}}{{{{\rm{t}}_{{\rm{1/2}}}}}}\)

As there are two isotopes, there is an expression for determining their activity:

\({\rm{activity = \lambda \times N}}\)

We may put the decay constant from above in that formula:

\({\rm{activity = }}\frac{{{\rm{In(2)}}}}{{{{\rm{t}}_{{\rm{1/2}}}}}}{\rm{ \times N}}\)

As a result, we may deduce that activity and half-life are inversely proportional:

\({\rm{activity \alpha }}\frac{{{\rm{In(2)}}}}{{{{\rm{t}}_{{\rm{1/2}}}}}}\)

If the half-life is longer, the activity will be lower. Uranium\({\rm{ - 232}}\) has a shorter half-life than uranium\({\rm{ - 233}}\), which means it has higher activity.

Therefore, the activity is said to be inversely proportional to half-life. It means that the activity will be smaller with a higher half-life. Uranium\({\rm{ - 232}}\) has to be a smaller half-life, and thus it has more activity than uranium\({\rm{ - 233}}\).