Question

The specific activity of a radioactive material is the number of disintegrations per second per gram of radioactive atoms. a) Given the half-life of \({ }^{14} \mathrm{C}\) of \(5730 \mathrm{yr}\), calculate the specific activity of \({ }^{14} \mathrm{C}\). Express your result in disintegrations per second per gram, becquerel per gram, and curie per gram. b) Calculate the initial activity of a \(5.00-\mathrm{g}\) piece of wood. c) How many \({ }^{14} \mathrm{C}\) disintegrations have occurred in a \(5.00-\mathrm{g}\) piece of wood that was cut from a tree January \(1,1700 ?\)

Short answer

Question: Calculate the specific activity of Carbon-14 and determine the initial activity of a 5-gram piece of wood. Also, find out how many Carbon-14 disintegrations have occurred in the same piece of wood since January 1, 1700. Answer: The specific activity of Carbon-14 is approximately [insert value from Step 2] Bq/g or [insert value from Step 3] Ci/g. The initial activity of the 5-gram piece of wood is approximately [insert value from Step 4] Bq. The number of Carbon-14 disintegrations that have occurred in the piece of wood since January 1, 1700, is approximately [insert value from Step 5] disintegrations.

Step-by-step solution

Calculate the decay constant for Carbon-14

To calculate the decay constant for Carbon-14, we can use the equation: \(\lambda = \frac{ln(2)}{t_{\frac{1}{2}}}\), where \(\lambda\) is the decay constant, and \(t_{\frac{1}{2}}\) is the half-life of the radioactive material. Given the half-life of Carbon-14 is 5730 years, we can calculate the decay constant as follows: \(\lambda = \frac{ln(2)}{5730\,\text{yr}}\)

Calculate the specific activity of Carbon-14

The specific activity of a radioactive material is given by: \(A = N \cdot \lambda\), where \(A\) is the specific activity, \(N\) is the number of radioactive atoms per gram, and \(\lambda\) is the decay constant calculated in Step 1. To find the number of atoms per gram for Carbon-14, we can use the equation: \(N = \frac{N_\text{A}}{M_\text{C}}\), where \(N_\text{A}\) is the Avogadro's number, and \(M_\text{C}\) is the molar mass of Carbon in grams per mole. Since the molar mass of Carbon-14 is 14 g/mol and the Avogadro's number is \(6.022 \times 10^{23}\,\text{mol}^{-1}\), we can calculate the number of atoms per gram as follows: \(N = \frac{6.022 \times 10^{23}}{14}\) Now, calculating the specific activity: \(A = N \cdot \lambda\)

Express specific activity in different units

The specific activity calculated in Step 2 is expressed in disintegrations per second per gram. To convert that value into becquerel per gram (Bq/g), we need to know that 1 Bq/g is equal to 1 disintegration per second per gram. Therefore, the specific activity in Bq/g is the same value. Next, to convert specific activity from becquerel per gram to curie per gram (Ci/g), we use the conversion factor: \({1 \mathrm{Ci}} = 3.7 \times 10^{10} \mathrm{Bq}\). Given the value calculated in the previous step, we just need to apply: \(A_\text{Ci/g} = \frac{A_\text{Bq/g}}{3.7 \times 10^{10}}\)

Calculate the initial activity of the piece of wood

Given the mass of the piece of wood (5 g), we can calculate the initial activity by multiplying the specific activity found in Step 2 by the mass. So, \(A_\text{initial} = A \cdot m\), where \(A_\text{initial}\) is the initial activity, \(A\) is the specific activity, and \(m\) is the mass of the piece of wood.

Calculate the number of Carbon-14 disintegrations since January 1, 1700

First, we need to calculate the elapsed time since January 1, 1700, to the present day. For simplicity, let's assume that the current date is January 1, 2022, which gives us a total of 322 years. To calculate the remaining Carbon-14 activity after the elapsed time, we can use the radioactive decay equation: \(A(t) = A_{0}\,e^{-\lambda t}\), where \(A(t)\) is the Carbon-14 activity at time \(t\), \(A_{0}\) is the initial activity, \(\lambda\) is the decay constant, and \(t\) is the elapsed time in years. First, convert the time to seconds for the equation: \(t_\text{s} = 322\,\text{yr} \times 365.25\, \text{days/yr} \times 24\, \text{hr/day} \times 60\, \text{min/hr} \times 60\, \text{s/min}\). Now, calculate the remaining Carbon-14 activity after 322 years: \(A(322\, \text{yr}) = A_\text{initial} \cdot e^{-\lambda \cdot t_\text{s}}\) Finally, we can find the number of disintegrations by calculating the difference in activity between the initial activity and the remaining activity: \(\Delta A = A_\text{initial} - A(322\, \text{yr})\) The number of Carbon-14 disintegrations that have occurred since January 1, 1700, in the 5.00 g piece of wood is given by \(\Delta A\).

Key concepts

Specific Activity of Radioactive Material

Radioactive materials emit radiation through a process known as radioactive decay, and understanding the rate at which this decay occurs is essential for various applications in science and medicine. The specific activity measures the intensity of radioactivity of a material by quantifying how many atomic nuclei decay per second in a given mass of the substance. It is expressed in units like disintegrations per second per gram, becquerels per gram (Bq/g), or curies per gram (Ci/g).

When we talk about specific activity, we are essentially referring to the ratio of activity (A) to the mass (m) of the radioactive material, where activity denotes the number of disintegrations per unit of time. The specific activity is related to the half-life and the total number of radioactive atoms present, and it can be calculated by the formula:
\[A = N \cdot \lambda\]
where \(N\) is the number of radioactive atoms per gram (determined from the molar mass and Avogadro's number) and \(\lambda\) is the decay constant (associated with the half-life of the substance).

• The decay constant (\(\lambda\)) represents the probability per unit time that an atom will decay. It can be calculated using the half-life (\(t_{1/2}\)) of the substance: \(\lambda = \frac{\ln(2)}{t_{1/2}}\).
• Knowing the value of \(\lambda\) and the number of atoms in a gram of the substance, the specific activity can be derived.
• Once the specific activity is found in disintegrations per second per gram, it can then be easily converted to other units like Bq/g (where 1 Bq = 1 disintegration per second), or to Ci/g using conversion factors.

For instance, in our exercise, to calculate the specific activity of Carbon-14, we first determine the decay constant using its half-life and then multiply it by the number of Carbon-14 atoms in one gram of the substance.

Half-Life

One of the fundamental concepts in radioactivity is half-life, which is the time it takes for half the atoms in a given sample of radioactive material to decay. The half-life is a crucial parameter that helps scientists determine how long a radioactive substance will remain active, and it is an intrinsic property of the radioactive isotope, not dependent on its amount or form.

Half-life is symbolized by the equation \(t_{1/2}\), and each radioactive isotope has a unique half-life that can range from fractions of seconds to millions of years. For instance, Carbon-14, used in radiocarbon dating, has a half-life of approximately 5730 years. This long half-life makes it possible for scientists to date ancient biological materials like fossils and archeological artifacts.

• The decay constant (\(\lambda\)) and the half-life are inversely related, and the concept of half-life allows us to understand that radioactive decay is exponential, not linear.
• Different isotopes have different half-lives, which affect how quickly they will become less radioactive over time.
• Half-life calculations are important in determining the proper dosage of radioactive substances in medical treatments, environmental testing, and nuclear waste management.

In the exercise, the half-life of Carbon-14 is a fundamental value needed to calculate the specific activity and to further understand the degree of disintegration over a period.

Carbon-14 Disintegration

Carbon-14 disintegration refers to the radioactive decay process of the Carbon-14 isotope, which is a natural occurrence in the environment. Carbon-14 is a rare version of carbon with two extra neutrons, making it unstable. Over time, it decays into nitrogen-14 through beta decay, and this process is what we track in radiocarbon dating.

During the disintegration process, Carbon-14 atoms in a sample will decrease exponentially, as described by the radioactive decay law. The formula that shows this relationship is: \[A(t) = A_0 e^{-\lambda t}\]
where \(A(t)\) is the activity at time \(t\), \(A_0\) is the initial activity, \(\lambda\) is the decay constant, and \(t\) is the elapsed time.

• The rate of disintegration helps scientists determine the age of carbon-containing materials by comparing the remaining Carbon-14 in a sample to what would have been present originally.
• Carbon-14 dating can provide an estimate of the age of organic material up to about 50,000 years old.
• Knowing the disintegration rate is fundamental in calculating the number of radioactive decays that have occurred over a period of time, as in part (c) of the exercise.

In part (c) of our exercise, understanding the disintegration of Carbon-14 is necessary to determine how much Carbon-14 has decayed since the piece of wood was cut in the year 1700, allowing us to estimate the number of disintegrations and the remaining activity in the sample.