Question

Bananas are somewhat radioactive due to the presence of substantial amounts of potassium. Potassium- 40 decays by two different paths: $$\begin{array}{l}_{19}^{40} \mathrm{K} \rightarrow_{20}^{40} \mathrm{Ca}+\beta^{-}(89.3 \%) \\ _{19}^{40} \mathrm{K} \rightarrow_{18}^{40} \mathrm{Ar}+\beta^{+}(10.7 \%)\end{array}$$ The half-life for potassium decay is \(1.3 \times 10^{9}\) years. Determine the rate constants for the individual channels.

Short answer

The rate constants for the individual decay channels are: - Rate constant for \(\beta^-\) decay: \( \lambda_{\beta^-} = 0.893 \cdot \frac{\ln{2}}{1.3 \times 10^9} \approx 4.68 \times 10^{-10} \, \text{years}^{-1} \) - Rate constant for \(\beta^+\) decay: \( \lambda_{\beta^+} = 0.107 \cdot \frac{\ln{2}}{1.3 \times 10^9} \approx 5.61 \times 10^{-11} \, \text{years}^{-1} \)

Step-by-step solution

Write down the given information.

We are given the following data: - Half-life for potassium decay: \( T_{1/2} = 1.3 \times 10^9 \) years - Decay percentages: - \(\beta^-\) decay: \( 89.3\% \) - \(\beta^+\) decay: \( 10.7\% \)

Convert percentages to fractions.

To work with the decay percentages, we need to convert them into fractions. We can do this by dividing the percentage by 100: - Fraction of \(\beta^-\) decay: \( 0.893 \) - Fraction of \( \beta^+ \) decay: \( 0.107 \)

Calculate the overall decay constant.

In order to find the individual rate constants, first we need to calculate the overall decay constant. The decay constant can be found using the half-life formula: \[ \lambda = \frac{\ln{2}}{T_{1/2}} \] where \(\lambda\) is the decay constant and \(T_{1/2}\) is the half-life. Plugging in the given half-life value, we get: \[ \lambda = \frac{\ln{2}}{1.3 \times 10^9} \]

Calculate the individual decay constants.

Now, we can find the individual rate constants for each decay path using the overall decay constant and the fractions. We simply multiply the overall decay constant \(\lambda\) by the fraction for each decay path: - Rate constant for \(\beta^-\) decay: \( \lambda_{\beta^-} = 0.893 \cdot \lambda \) - Rate constant for \(\beta^+\) decay: \( \lambda_{\beta^+} = 0.107 \cdot \lambda \)

Evaluate the individual decay constants.

Now, plug in the value for the overall decay constant \(\lambda\) to get the final rate constants for the individual channels: - Rate constant for \(\beta^-\) decay: \( \lambda_{\beta^-} = 0.893 \cdot \frac{\ln{2}}{1.3 \times 10^9} \) - Rate constant for \(\beta^+\) decay: \( \lambda_{\beta^+} = 0.107 \cdot \frac{\ln{2}}{1.3 \times 10^9} \) Simplify these expressions to find the individual decay constants for the two decay paths.

Key concepts

Potassium-40 Decay

Potassium-40 \(_{19}^{40}K\) is a naturally occurring isotope of potassium that is radioactive. This means it undergoes decay over time, releasing radiation in the process. Potassium-40 is found in many things around us, including bananas, which is why people sometimes joke that bananas are radioactive.The potassium-40 decay process is unique because it can follow two different decay paths:

• One path leads to the formation of calcium-40 \(_{20}^{40}Ca\) through beta-minus (\(\beta^-\)) decay.
• The second path leads to the formation of argon-40 \(_{18}^{40}Ar\) through beta-plus (\(\beta^+\)) decay.

The distinct pathways allow potassium-40 to transform into different elements, which impacts fields ranging from nutrition to geologic studies. Understanding these decay paths helps scientists in dating ancient geological events using radiometric dating methods.

Decay Constant

The decay constant \( \lambda \) is a fundamental concept in radioactive decay that tells us how quickly an element like potassium-40 decays over time. Mathematically, it represents the probability per unit time that a given atom will decay.To find the decay constant for potassium-40, we use the formula:\[\lambda = \frac{\ln{2}}{T_{1/2}}\]where \(T_{1/2}\) is the half-life of the isotope. This formula helps to connect the half-life and the decay rate, simplifying calculations related to radioactive substances.Once the overall decay constant is known, it is possible to determine the rate constants for individual decay channels by multiplying \( \lambda \) by the fraction of decay occurring through each channel. This makes the decay constant an integral part of understanding radioactive processes, serving as a key variable in multiple scientific and practical applications.

Half-life Calculation

The half-life of a radioactive isotope, like potassium-40, is the time required for half of the isotopic atoms in a sample to undergo decay. This is a crucial measurement because it gives insights into the stability and lifespan of the isotope.For potassium-40, the half-life is \( 1.3 \times 10^9 \) years. This remarkably long half-life demonstrates the slow rate of decay for this isotope, which is why significant amounts still exist naturally.The half-life transforms easily into a decay constant, helping calculate how quickly a sample will reduce to half its initial quantity. More specifically, once the half-life is known, the decay constant can be computed using:

• \( \lambda = \frac{\ln{2}}{T_{1/2}} \)

This enables scientists to predict how long a radioactive sample will remain active, forming the basis for dating archaeological finds and understanding biological implications of elements like potassium in diet.

Beta Decay Channels

In the decay of potassium-40, there are two principal beta decay channels: beta-minus (\(\beta^-\)) decay and beta-plus (\(\beta^+\)) decay.1. **Beta-minus (\(\beta^-\)) Decay:** - In this process, a neutron inside the potassium-40 nucleus is transformed into a proton. This leads to the emission of an electron (beta particle) and an antineutrino. - The decay results in the formation of calcium-40 \(_{20}^{40}Ca\), contributing to the majority (\(89.3\%\)) of potassium-40 decay.2. **Beta-plus (\(\beta^+\)) Decay:** - Here, a proton is converted into a neutron, which causes the emission of a positron (positive beta particle) and a neutrino. - This decay pathway produces argon-40 \(_{18}^{40}Ar\), and it accounts for a smaller portion (\(10.7\%\)) of decays.Understanding these channels helps in calculating individual decay constants for each path by multiplying the overall decay constant by their respective fractions. This knowledge is instrumental in multiple scientific investigations, including dating rocks and tracing geochemical pathways.