Question

It took \(49.2 \mathrm{~s}\) for \(3.000 \mathrm{~g}\) of a radioactive isotope to decay to \(0.1875 \mathrm{~g}\). What is the half-life of this isotope?

Short answer

The half-life of the radioactive isotope is approximately 12.29 seconds.

Step-by-step solution

Understand the Decay Process

The problem involves radioactive decay, which follows a predictable pattern. The formula for radioactive decay is given by \( N(t) = N_0 \, e^{-\lambda t} \), where \( N(t) \) is the amount remaining at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the decay constant. The half-life \( T_{1/2} \) is the time it takes for half of the material to decay.

Identify the Known Values

From the exercise, we know that the initial mass \( N_0 = 3.000 \) g and the remaining mass \( N(t) = 0.1875 \) g after \( t = 49.2 \) seconds.

Use the Decay Formula

Substitute the known values into the decay formula: \[ 0.1875 = 3.000 \times e^{-\lambda \times 49.2} \]Solving for \( e^{-\lambda \times 49.2} \), we first divide both sides by 3.000: \[ \frac{0.1875}{3.000} = e^{-\lambda \times 49.2} \] \[ 0.0625 = e^{-\lambda \times 49.2} \].

Solve for Decay Constant \( \lambda \)

Take the natural logarithm (ln) of both sides to solve for \( \lambda \):\[ \ln(0.0625) = -\lambda \times 49.2 \]. Calculate \( \ln(0.0625) \) which is approximately -2.7726. Solve for \( \lambda \):\[ \lambda = \frac{-2.7726}{-49.2} \approx 0.0564 \text{ s}^{-1} \].

Calculate the Half-Life

The half-life \( T_{1/2} \) is related to the decay constant by the formula:\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]. Substitute the value of \( \lambda \) into the equation:\[ T_{1/2} = \frac{0.693}{0.0564} \approx 12.29 \text{ seconds} \].

Key concepts

Radioactive Decay

In the world of physics, radioactive decay is a fascinating process where unstable atomic nuclei lose energy by emitting radiation. This transformation happens at a predictable rate, which can be understood through mathematical equations. Here’s how it works:

• Radioactive decay is a random process on the atomic level. However, when observing a large number of atoms, the decay follows a specific pattern that can be described by the decay formula.
• The fundamental formula for radioactive decay is expressed as \( N(t) = N_0 \, e^{-\lambda t} \), where:
  • \(N(t)\) represents the quantity of the substance that still remains after time \(t\).
  • \(N_0\) stands for the substance's initial quantity.
  • \(\lambda\) is known as the decay constant, an indicator of the rate at which the element decays.
• This equation shows that the amount of radioactive isotope decreases exponentially over time.

Understanding these principles helps us calculate essential parameters like the half-life of a substance, which is crucial for various practical applications.

Decay Constant

The decay constant, represented by the symbol \( \lambda \), is of utmost importance in the study of radioactive decay. It quantifies the probability per unit time that a given atom will decay. Here's a closer look:

• The decay constant is a proportionality factor that provides a connection between the decay rate and the amount of substance present at any given time.
• When given, the decay constant lets us predict how quickly a radioactive isotope will diminish over time.
• In calculations, using the natural logarithm simplifies the process. By taking the natural log of the decay equation, we can solve for \( \lambda \) when other variables are known, such as \( \ln(0.0625) = -\lambda \times 49.2 \).
• Pretty useful! Once we have \( \lambda \), we can easily find the half-life, which tells us how long it takes for half of the material to decay.

Remember, a smaller \( \lambda \) indicates a slower decay process, while a larger value means the substance decays more quickly.

Natural Logarithm

The natural logarithm, often abbreviated as "ln," is an essential mathematical concept employed in exponential decay calculations, including radioactive decay. Here's why it’s so crucial:

• The natural logarithm is the inverse function of the exponential function \( e^x \). This property allows it to be particularly useful in solving equations involving exponentials.
• In the context of radioactive decay, the natural logarithm helps isolate variables, such as the decay constant \( \lambda \), from exponential equations. For example, \( \ln(0.0625) = -\lambda \times 49.2 \).
• Arithmetic with natural logs can transform complex multiplicative relationships into simpler additive ones, facilitating straightforward calculations.
• Understanding the relationship between exponential growth/decay and natural logarithms is key to mastering decay processes in nuclear physics.

So, whenever you're dealing with exponential decay, remember that the natural logarithm is your friend in untangling those complex equations!

Radioactive Isotope

A radioactive isotope, or radioisotope, refers to an isotope whose nucleus is unstable. These are atoms that undergo radioactive decay, emitting radiation in the process. Here’s what you need to know:

• Every element can have numerous isotopes, distinguished by the number of neutrons in their nuclei.
• Some isotopes are stable, while others are radioactive, meaning they emit radiation as they decay into a more stable form.
• The rate at which this decay happens is characterized by the decay constant and the substance's half-life.
• Radioactive isotopes have wide-ranging applications, from medical imaging and cancer treatments to archaeological dating and nuclear energy.

Understanding the nature of radioactive isotopes enhances our ability to harness nuclear energy safely and effectively, among other uses.