Question

A freshly prepared sample of a radioisotope of half-life \(1386 \mathrm{~s}\) has activity \(10^{3}\) disintegrations per second. Given that \(\ln 2=0.693\), the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first \(80 \mathrm{~s}\) after preparation of the sample is ____ .

Short answer

The nearest integer percentage of decayed nuclei in the first 80 seconds is 5%.

Step-by-step solution

- Understanding the decay process

The decay of a radioisotope is described by its half-life, which is the time required for the number of nuclei to decrease to half its initial value. The fraction of the initial number of nuclei that decay over a time period can be determined by using the decay law, represented by the equation: \( N(t) = N_0 e^{-kt} \), where \(N(t)\) is the number of undecayed nuclei at time \(t\), \(N_0\) is the initial number of undecayed nuclei, \(k\) is the decay constant, and \(e\) is the base of the natural logarithm.

- Determining the decay constant

To calculate the decay constant \(k\), use the relationship between the half-life \(T_{1/2}\) and the decay constant: \(k = \frac{\ln 2}{T_{1/2}}\). Given \(T_{1/2} = 1386\) seconds and \(\ln 2 = 0.693\), we find \(k = \frac{0.693}{1386}\).

- Calculating the fraction of decayed nuclei

With the decay constant \(k\), we can apply the decay law to calculate the fraction of nuclei that decay in the first 80 seconds. \( N(80) = N_0 e^{-k \cdot 80} \). The fraction of decayed nuclei is then \( 1 - \frac{N(80)}{N_0} \).

- Substituting the values

Substitute the values \(k\) and \(t=80\) seconds into the equation from Step 3 to get the fraction of decayed nuclei. After finding this value, convert the fraction to a percentage and round it to the nearest integer.

- Calculating the result

Performing the calculation from Step 4 yields \( 1 - e^{-k \cdot 80} \), where \( e^{-k \cdot 80} \) is the remaining fraction of undecayed nuclei after 80 seconds. After calculating this, subtract from 1 to find the decayed fraction and round to nearest percentage.

Key concepts

Decay Constant

Understanding the decay constant is essential when studying radioisotope decay. In nuclear physics, the decay constant, often represented by the symbol \(k\), quantifies the probability per unit time that a given nucleus will decay. It is a fundamental parameter in the decay law equation \( N(t) = N_0 e^{-kt} \), where \(N(t)\) is the number of undecayed nuclei at time \(t\), and \(N_0\) is the initial number of undecayed nuclei.

The decay constant is inversely proportional to the half-life (\(T_{1/2}\)) of the radioisotope, which can be expressed by the formula: \( k = \frac{\ln 2}{T_{1/2}} \). The logarithm base \(e\) is a mathematical constant approximately equal to 2.71828. In the context of the given exercise, knowing the half-life of the radioisotope, we use the relationship between half-life and decay constant to calculate \(k\), which then allows us to predict how quickly the substance will decay over time.

Half-life

The concept of half-life is a cornerstone of nuclear physics and radioisotope decay calculations. The half-life \(T_{1/2}\) is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. This is a characteristic property of each radioactive isotope and can range from fractions of a second to millions of years, depending on the isotope in question.

Mathematically, the half-life is tied to the decay constant as mentioned earlier. This relationship forms the basis for solving numerous problems in radioactivity, including calculating the age of archaeological finds or the dosage in medical treatments. For example, in our exercise, knowing the half-life is crucial to determining the fraction of the radioactive material that will decay in a specific time frame, by enabling us to find the decay constant first.

Nuclear Physics

Nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. The principles of nuclear physics apply directly to the understanding of radioisotopes and their decay patterns. At its core, this discipline attempts to explain the processes that power the sun, the nature of nuclear force, and the mechanics behind elements transforming into other elements through radioactive decay.

Radioactive decay itself is a random process at the level of single atoms, described statistically by equations learnt in nuclear physics. It is this statistical nature that allows us to predict the behavior of a large number of atoms, despite not being able to predict individual events. In the context of our exercise, nuclear physics provides the theoretical framework that explains why certain materials emit radiation over time and how this can be quantified with precise mathematical relationships, such as the decay law.