Question

A radioactive sample has an initial activity of 28 dpm. Half hour later, the activity is \(14 \mathrm{dpm}\). How many atoms of the radioactive nuclide were there originally? \((\ln 2=0.7)\) (a) 1200 (b) 200 (c) 600 (d) 300

Short answer

The original number of atoms was 1200.

Step-by-step solution

Understand the Concept of Half-Life

The activity of a radioactive sample is proportional to the number of radioactive atoms present. Activity is halved when one half-life passes. Since the activity goes from 28 dpm to 14 dpm in half an hour, we can conclude that the half-life of this nuclide is 30 minutes.

Calculate the Decay Constant

The decay constant, denoted by \(\lambda\), is related to the half-life (\(T_{1/2}\)) by the equation \(\lambda = \frac{\ln 2}{T_{1/2}}\). Here, we use the given value of \(\(\ln 2 = 0.7\)\) and the half-life of 30 minutes to calculate the decay constant \(\lambda\).

Calculate the Number of Original Atoms

Using the decay constant, we can find the initial number of atoms (\(N_0\)) by the equation relating activity (\(A\)) and the number of atoms: \(A = \lambda N\). Since the initial activity (\(A_0\)) is 28 dpm, we can rearrange the formula to \(N_0 = \frac{A_0}{\lambda}\) and substitute the known values to calculate \(N_0\).

Calculate Initial Number of Atoms Using Known Values

First, find the decay constant using the half-life. We have \(\lambda = \frac{0.7}{30}\) minutes\(^{-1}\). Next, calculate the number of atoms using \(N_0 = \frac{28}{\lambda}\) which gives us the initial number of radioactive nuclide atoms.

Key concepts

Half-Life Calculation

Understanding the 'half-life' of a radioactive substance is essential when studying nuclear chemistry. It is the time required for half of the radioactive atoms in a sample to decay. From the given problem, we observe that the activity of the radioactive sample decreased from 28 disintegrations per minute (dpm) to 14 dpm in 30 minutes. This indicates that the half-life of the nuclide in question is indeed 30 minutes.

The calculation of half-life can sometimes be more complex when not directly observable through simple halving of activity. However, with a known decay constant, the half-life can be algebraically deduced from the relationship \( T_{1\/2} = \frac{\ln 2}{\lambda} \). In the context of our exercise, the half-life was not calculated as such because it was directly given, but understanding this relationship is crucial for more complex problems.

Decay Constant

The 'decay constant' (denoted by \(\lambda\)) is a probability factor that describes the likelihood of the decay of a single atom in a given time interval. In other words, it is a measurement of the rate at which a radioactive substance decays. To find the decay constant using the known half-life, we use the equation \(\lambda = \frac{\ln 2}{T_{1\/2}}\), where \(\ln 2\) is the natural logarithm of 2, approximately equal to 0.7.

By substituting the half-life, 30 minutes in this case, we can calculate the decay constant necessary for further calculations involving the initial activity and the number of atoms in the sample. Understanding this fundamental parameter is key to linking the concepts of half-life and overall decay processes.

Activity of Radioactive Sample

The 'activity' of a radioactive sample is a measure of the rate at which the atoms within the sample are decaying. It's usually expressed in units such as disintegrations per minute (dpm) or becquerels. Activity is directly proportional to the number of undecayed radioactive atoms present in the sample. The initial activity, therefore, can provide essential information about the quantity of radioactive material.

Utilizing the decay constant and the equation \(A = \lambda N\), where \(A\) is the activity and \(N\) is the number of atoms, students can solve for the original number of atoms in the radioactive sample by rearranging it to \(N_0 = \frac{A_0}{\lambda}\). Using this approach in the given context, we derived the initial quantity of radioactive nuclide atoms, ensuring that students not only follow the calculation but also comprehend the principles that define radioactive decay processes.