Question

The mass percent of carbon in a typical human is \(18 \%\), and the mass percent of \({ }^{14} \mathrm{C}\) in natural carbon is \(1.6 \times 10^{-10} \%\). Assuming a \(180-\mathrm{lb}\) person, how many decay events per second occur in this person due exclusively to the \(\beta\) -particle decay of \({ }^{14} \mathrm{C}\) (for \({ }^{14} \mathrm{C}\), \(t_{1 / 2}=5730\) years)?

Short answer

In a 180-pound person, approximately \(1.95 \times 10^{12}\) decay events per second occur due exclusively to the beta-particle decay of Carbon-14.

Step-by-step solution

1. Determine the mass of carbon in the person

We are given that the mass percent of carbon in a typical human is 18%. A 180-pound person would have this mass in kg: \( 180 ~\text{lb} \cdot \frac{1 ~\text{kg}}{2.2046 ~\text{lb}} \approx 81.65 ~\text{kg}\) The mass of carbon in this person is: \( \text{mass of carbon} = 81.65 ~\text{kg} \cdot 18 \% = 14.697 ~ \text{kg} \)

2. Determine the number of Carbon-14 atoms in the person

We are given that the mass percent of Carbon-14 in natural carbon is 1.6x10^(-10)%. The mass of Carbon-14 in this person would be: \( \text{mass of Carbon-14} = 14.697 ~\text{kg} \cdot 1.6 \times 10^{-10} \% \approx 2.35152 \times 10^{-8} ~ \text{kg} \) Now we can find the number of Carbon-14 atoms by dividing the mass of Carbon-14 by its atomic mass: \( \text{Number of Carbon-14 atoms} = \frac{\text{mass of Carbon-14}}{\text{atomic mass of Carbon-14}} \) where atomic mass of Carbon-14 is approximately 14 atomic mass unit(AMU): \( 1 ~\text{AMU} \approx 1.66 \times 10^{-27} ~\text{kg} \) Hence, \( \text{Number of Carbon-14 atoms} = \frac{2.35152 \times 10^{-8} ~\text{kg}}{14 \cdot 1.66 \times 10^{-27} ~\text{kg}} \approx 5.09008 \times 10^{23} ~\text{atoms}\)

3. Calculate the decay constant of Carbon-14

The decay constant (λ) can be calculated from the half-life (t_(1/2)) using the following formula: \( \lambda = \frac{\ln 2}{t_{1/2}} \) Given that the half-life of Carbon-14 is 5730 years, we first convert it to seconds: \( t_{1/2} = 5730 ~\text{years} \cdot \frac{365.25 ~\text{days}}{1 ~\text{year}} \cdot \frac{24 ~\text{hours}}{1 ~\text{day}} \cdot \frac{60 ~\text{mins}}{1 ~\text{hour}} \cdot \frac{60 ~\text{seconds}}{1 ~\text{min}} \approx 1.8075 \times 10^{11} ~ \text{seconds} \) Now, we can calculate the decay constant: \( \lambda \approx \frac{\ln 2}{1.8075 \times 10^{11} ~\text{seconds}} \approx 3.83465 \times 10^{-12} ~\text{s}^{-1}\)

4. Calculate the number of decay events per second

The number of decay events per second (decays per second) can be calculated by multiplying the number of Carbon-14 atoms by the decay constant: \( \text{Decays per second} = \text{Number of Carbon-14 atoms} \cdot \lambda \) \( \text{Decays per second} \approx 5.09008 \times 10^{23} ~\text{atoms} \cdot 3.83465 \times 10^{-12} ~\text{s}^{-1} \approx 1.95 \times 10^{12} ~\text{decays/s}\) So, there are approximately \(1.95 \times 10^{12}\) decay events per second in a 180-pound person due exclusively to the beta-particle decay of Carbon-14.

Key concepts

Mass Percent Calculation

The mass percent calculation is an important concept in chemistry, especially when determining the amount of a certain element in a compound. In this context, we are looking at the mass percent of carbon in a typical human and how it relates to the calculation of Carbon-14.
We start with the given mass percent of carbon in a person, which is 18%. This tells us that 18% of a person's body mass is carbon. To find out the mass of carbon in a person weighing 180 pounds, we begin by converting pounds to kilograms, since the standard unit for mass calculation is kilograms.

• The conversion factor is: 1 lb = 0.453592 kg, thus 180 lb is approximately 81.65 kg.
• Now, to find the mass of carbon, we use the formula: \[ \text{Mass of carbon} = \text{Total mass} \times \left( \frac{\text{Percent of carbon}}{100} \right) \]
• Plugging in the values: \[ \text{Mass of carbon} = 81.65 \text{ kg} \times 0.18 = 14.697 \text{ kg} \]

This gives us the mass of carbon in the person's body, which is crucial for further calculations involving isotopes like Carbon-14.

Isotope Decay Calculation

Isotope decay calculation helps determine how an isotope like Carbon-14 decays over time within a given sample, such as in a human body.
Knowing the mass percent of Carbon-14 in natural carbon helps in calculating its presence in the human body. It is given as 1.6 x 10^-10 %, which means only a tiny fraction of the carbon in a person's body is Carbon-14. To find the actual mass of Carbon-14, we use the given mass of carbon in the person:

• Calculate the mass of Carbon-14: \[ \text{Mass of Carbon-14} = 14.697 \text{ kg} \times \left(\frac{1.6 \times 10^{-10}}{100}\right) \]
• Which simplifies to approximately 2.35152 x 10^-8 kg.

Next, we convert this mass into the number of Carbon-14 atoms:

• Use the formula: \[ \text{Number of Carbon-14 atoms} = \frac{\text{mass of Carbon-14}}{\text{atomic mass of Carbon-14}} \]
• Where the atomic mass of Carbon-14 is approximately 14 AMU, and 1 AMU is 1.66 x 10^-27 kg.
• So, we calculate: \[ \frac{2.35152 \times 10^{-8} \text{ kg}}{14 \times 1.66 \times 10^{-27} \text{ kg}} \approx 5.09008 \times 10^{23} \text{ atoms} \]

This large number represents how many Carbon-14 atoms are present in the 180-pound person, setting the stage for calculating decay events.

Decay Constant

The decay constant is a important parameter in determining how quickly a radioactive isotope decays. It is particularly useful when dealing with isotopes like Carbon-14, which has a known half-life.
The decay constant, denoted by \(\lambda\), is calculated using the half-life of the isotope. For Carbon-14, the half-life is 5730 years. We convert this into seconds to match the units used in the decay calculations. First, we calculate the half-life in seconds:

• 1 year has 365.25 days to account for leap years.
• The conversion process is as follows: \[ t_{1/2} = 5730 \text{ years} \times \frac{365.25 \text{ days}}{1 \text{ year}} \times \frac{24 \text{ hours}}{1 \text{ day}} \times \frac{60 \text{ mins}}{1 \text{ hour}} \times \frac{60 \text{ seconds}}{1 \text{min}} \approx 1.8075 \times 10^{11} \text{ seconds} \]

Now, we find the decay constant:

• The formula to use is: \[ \lambda = \frac{\ln 2}{t_{1/2}} \]
• By substituting the values: \[ \lambda \approx \frac{\ln 2}{1.8075 \times 10^{11} \text{ seconds}} \approx 3.83465 \times 10^{-12} \text{s}^{-1} \]

This decay constant is then used to calculate how many atoms decay per second. By multiplying the decay constant with the number of Carbon-14 atoms, we discover the activity level, or decay events per second, crucial for understanding the radioactive behavior.