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Magazine Chapter 19: Problem 31
The initial radioactivity of a cobalt- 60 sample was 1200 dpm and after 21.2 years the activity dropped to \(75 \mathrm{dpm}\). What is the half-life of the radionuclide?
Short Answer
Expert verified
The half-life of cobalt-60 is approximately 5.30 years.
Step by step solution
01
Understand the Decay Formula
The decay of a radioactive substance is described by the formula \( A = A_0 \times e^{-kt} \), where \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( k \) is the decay constant, and \( e \) is the base of natural logarithms.
02
Identify Known Values
From the problem, we know that \( A_0 = 1200 \text{ dpm} \), \( A = 75 \text{ dpm} \), and \( t = 21.2 \text{ years} \). Our goal is to find the half-life (\( t_{1/2} \)) of the radionuclide.
03
Solve for Decay Constant (k)
Rearrange the decay formula to solve for \( k \): \( k = -\frac{1}{t} \ln\left(\frac{A}{A_0}\right) \). Plug in the known values: \( k = -\frac{1}{21.2} \ln\left(\frac{75}{1200}\right) \). Calculate the natural logarithm and the decay constant.
04
Calculate the Decay Constant
First, calculate \( \ln\left(\frac{75}{1200}\right) = \ln(0.0625) \approx -2.7726 \). Then, \( k = -\frac{1}{21.2} \times (-2.7726) \approx 0.13075 \text{ per year} \).
05
Use Formula for Half-life
The half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{0.693}{k} \). Substitute the value of \( k \) into this formula to find the half-life.
06
Calculate Half-life
Substitute \( k \approx 0.13075 \) into the half-life formula: \( t_{1/2} = \frac{0.693}{0.13075} \approx 5.30 \text{ years} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radioactive Decay
Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process results in the transformation of the original atom into a different element or a different isotope of the same element. Radioactive decay occurs at the atomic level and can happen through several different pathways, including alpha decay, beta decay, and gamma decay.
The rate at which a radioactive substance decays is described by specific mathematical equations. It is predictable and constant for a given substance, making it a powerful tool for dating geological and archaeological samples.
- Alpha decay: Involves the emission of an alpha particle (2 protons and 2 neutrons), reducing the atomic number by 2 and the mass number by 4.
- Beta decay: Involves the transformation of a neutron into a proton (or vice versa), emitting a beta particle (an electron or positron).
- Gamma decay: Involves the release of gamma photons, occurring when the nucleus transitions from a higher energy state to a lower one without changing the element's identity.
The rate at which a radioactive substance decays is described by specific mathematical equations. It is predictable and constant for a given substance, making it a powerful tool for dating geological and archaeological samples.
Decay Constant
The decay constant, represented by the symbol \( k \), is a key term in the formula used to calculate radioactive decay. It defines the probability per unit time that a nucleus will decay. Specifically, the decay constant is crucial for determining how fast a radioactive substance will decrease over time.
Mathematically, the decay constant \( k \) is related to the half-life \( t_{1/2} \) by the equation: \( t_{1/2} = \frac{0.693}{k} \). This relationship shows how these two parameters are interconnected and helps to calculate the half-life if \( k \) is known.
- The decay constant is unique for each radioactive isotope and is related to its half-life.
- A larger decay constant means the substance will decay more quickly.
- The decay constant is measured in units of inverse time (e.g., per year, per second).
Mathematically, the decay constant \( k \) is related to the half-life \( t_{1/2} \) by the equation: \( t_{1/2} = \frac{0.693}{k} \). This relationship shows how these two parameters are interconnected and helps to calculate the half-life if \( k \) is known.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a mathematical function that is the inverse of the exponential function \( e^x \). Understanding natural logarithms is crucial when dealing with the exponential decay equations in radioactive decay calculations.
In the context of the given problem, the natural logarithm of the ratio of final to initial activity helped determine the decay constant \( k \). This is a common step in solving decay-related problems.
- The natural logarithm has a base \( e \), which is approximately equal to 2.71828.
- It is used to linearize exponential growth or decay, making calculations more manageable.
- In radioactive decay equations, the natural logarithm helps to solve for variables like the decay constant or the amount of substance remaining after a period of time.
In the context of the given problem, the natural logarithm of the ratio of final to initial activity helped determine the decay constant \( k \). This is a common step in solving decay-related problems.
Cobalt-60
Cobalt-60 is a radioactive isotope of cobalt, often used in medical applications and industrial radiography. It emits gamma rays, which makes it highly useful for these purposes. Cobalt-60 has a known half-life, making it a common subject in half-life and decay calculations.
Understanding the properties and decay of Cobalt-60 contributes significantly to various applications in scientific and medical fields, and illustrates the principles of radioactive decay in a practical manner.
- Cobalt-60 undergoes beta decay to become an isotope of nickel, with the emission of beta particles and gamma rays.
- Its decay is described using the same mathematical principles applicable to all radioactive substances.
- The energy released in Cobalt-60 decay is harnessed in medical treatments like radiotherapy for cancer and sterilization of medical equipment.
Understanding the properties and decay of Cobalt-60 contributes significantly to various applications in scientific and medical fields, and illustrates the principles of radioactive decay in a practical manner.
