Question

The number of radioactive nuclides in a sample decays from \(1.00 \times 10^{20}\) to \(2.50 \times 10^{19}\) in 10.0 minutes. What is the half-life of this radioactive species?

Short answer

The half-life of this radioactive species is approximately 10.0 minutes.

Step-by-step solution

Understand the radioactive decay formula

The radioactive decay formula is given by: \[N(t) = N_0 \cdot e^{-\lambda t}\] where \(N(t)\) is the quantity of radioactive nuclides remaining after time \(t\), \(N_0\) is the initial number of radioactive nuclides, \(\lambda\) is the decay constant, and \(t\) is time.

Use the given data to find the decay constant

We have been given the initial number of radioactive nuclides \(N_0 = 1.00 \times 10^{20}\), the final number of nuclides after the decay \(N(t) = 2.50 \times 10^{19}\), and the time interval during which this decay occurs \(t = 10.0 \, \text{minutes}\). Plugging these values into the radioactive decay formula, we get: \[2.50 \times 10^{19} = (1.00 \times 10^{20}) \cdot e^{-\lambda (10.0 \, \text{min})}\] Let's isolate \(\lambda\) on one side of the equation.

Isolate the decay constant

To do this, first divide both sides of the equation by \(1.00 \times 10^{20}\): \[\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}} = e^{-\lambda (10.0 \, \text{min})}\] Next, take the natural logarithm of both sides. \[\ln{\left(\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}}\right)} = -\lambda (10.0 \, \text{min})\] Now, divide by \(-10.0 \, \text{min}\) to isolate \(\lambda\): \[\lambda = -\frac{\ln{\left(\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}}\right)}}{10.0 \, \text{min}} \approx 0.0693 \, \text{min}^{-1}\] We found the decay constant, \(\lambda \approx 0.0693 \, \text{min}^{-1}\).

Find half-life using the decay constant

The half-life of a radioactive species is related to the decay constant by the formula: \[T_{1/2} = \frac{\ln{2}}{\lambda}\] We can now substitute the value of \(\lambda\) that we found in the previous step and compute the half-life: \[T_{1/2} = \frac{\ln{2}}{0.0693 \, \text{min}^{-1}} \approx 10.0 \, \text{minutes}\] Therefore, the half-life of this radioactive species is approximately 10.0 minutes.

Key concepts

Half-Life Calculation

The concept of half-life is a fundamental part of understanding how radioactive materials decay over time. Half-life refers to the time required for half of the radioactive nuclides in a sample to decay. It's crucial for estimating how long a radioactive sample will remain active.

In this scenario, to calculate the half-life, we start by determining the decay constant \( \lambda \) using the decay formula: \[ N(t) = N_0 \cdot e^{-\lambda t} \] Here, \( N_0 \) represents the initial number of nuclides, \( N(t) \) the number remaining after time \( t \), and \( \lambda \) is the decay constant.

Once the decay constant is known, it can directly be applied in the formula \( T_{1/2} = \frac{\ln{2}}{\lambda} \) to find the half-life. This relationship indicates that the half-life is inversely proportional to the decay constant, meaning the faster something decays, the shorter its half-life.

Decay Constant

The decay constant \( \lambda \) is a key component in the study of radioactive decay. It quantifies the rate at which a radioactive isotope decays. A higher decay constant indicates a faster rate of decay.

In calculating the decay constant, we rearrange the standard decay formula to solve for \( \lambda \) by substituting in known values, such as the initial and remaining nuclide counts, and the elapsed time.

• Start by expressing the proportion of remaining nuclides using \( \frac{N(t)}{N_0} \).
• Apply the natural logarithm to simplify the decay equation.
• Finally, solve for \( \lambda \) in terms of these quantities.

Using this constant, we can predict the behavior over time of radioactive substances, which is crucial for many applications in science and industry.

Radioactive Nuclides

Radioactive nuclides are unstable atoms that seek stability by releasing energy in the form of radiation. This process alters the original nuclide into a different element or isotope. The nature and amount of radiation depend on the specific nuclide involved.

Radioactive decay is an unpredictable process on the level of individual atoms but is statistically predictable across large quantities. Thus, understanding the population dynamics of radioactive nuclides involves:

• Initial number of nuclides, represented by \( N_0 \).
• The reduction in number over a set time span, indicating how many atoms have decayed.
• Use of mathematical models like the decay formula to predict future states.

Studying these patterns helps in applications ranging from medical imaging to dating ancient artifacts. Through these processes, the behavior and quantity of radioactive nuclides can be tracked effectively over time.