Problem 47 - Chemistry: Principles and Reactions

Question

Argon- 41 is used to measure the rate of gas flow. It has a decay constant of \(6.3 \times 10^{-3} \mathrm{~min}^{-1}\). (a) What is its half-life? (b) How long will it take before only \(1.00 \%\) of the original amount of Ar- 41 is left?

Step-by-Step Solution

Answer

**Short Answer**: The half-life of Argon-41 is approximately 110.16 minutes. It will take approximately 460.24 minutes for only 1% of the original amount of Argon-41 to be left remaining.

Step 1: Calculate the half-life of Argon-41

To find the half-life of Argon-41, we will use the equation \(t_{1/2} = \frac{\ln 2}{\lambda}\). We are given the decay constant \(\lambda = 6.3 \times 10^{-3} \mathrm{~min}^{-1}\). Plugging this into the equation, we get: \(t_{1/2} = \frac{\ln 2}{6.3 \times 10^{-3} \mathrm{~min}^{-1}}\) Calculate the value of \(t_{1/2}\).

Step 2: Find the time required for only 1% of the original Argon-41 to be left

We now want to find the time \(t\) when only 1% of the original amount of Argon-41 is left. We can use the decay formula \(N_t = N_0 e^{-\lambda t}\). Since we are only concerned with the percentage of remaining Argon-41, we can set \(N_0 = 100\) and \(N_t = 1\). So, the equation becomes: \(1 = 100 e^{-6.3 \times 10^{-3} \mathrm{~min}^{-1} t}\) To solve for \(t\), we can first divide both sides by 100: \(\frac{1}{100} = e^{-6.3 \times 10^{-3} \mathrm{~min}^{-1} t}\) Next, take the natural logarithm of both sides: \(\ln(\frac{1}{100}) = -6.3 \times 10^{-3} \mathrm{~min}^{-1} t\) Now, divide both sides by the decay constant to isolate \(t\): \(t = \frac{\ln(\frac{1}{100})}{-6.3 \times 10^{-3} \mathrm{~min}^{-1}}\) Finally, calculate the value of \(t\).

Key Concepts

Half-Life Calculation

Half-life is a critical concept in understanding radioactive decay. It is the time required for half of the radioactive isotopes in a sample to decay. This means that after one half-life, 50% of the original material remains. To calculate the half-life, we use the formula \[ t_{1/2} = \frac{\ln 2}{\lambda} \]where \( t_{1/2} \) is the half-life and \( \lambda \) is the decay constant.
In the case of Argon-41, which has a decay constant of \( 6.3 \times 10^{-3} \mathrm{~min}^{-1} \), we can find the half-life by inserting the value into the formula:
- \( t_{1/2} = \frac{\ln 2}{6.3 \times 10^{-3} \mathrm{~min}^{-1}} \)
By performing these calculations, you will discover it takes a certain number of minutes for half of the Argon-41 to decay. Understanding and calculating half-life helps in determining how isotopes like Argon-41 are used in measuring and experimenting with decay rates.

Decay Constant

The decay constant is a crucial part of understanding radioactive decay. It represents the probability per unit time that a single atom will decay. This constant is an integral part of decay equations and helps determine how quickly a sample of radioactive material will reduce in quantity.

• The decay constant is symbolized by \( \lambda \).
• It is expressed in terms of inverse time units, such as \( \mathrm{min}^{-1} \).

For Argon-41, the decay constant is \( 6.3 \times 10^{-3} \mathrm{~min}^{-1} \). This value informs us about the rate at which Argon-41 decays. The higher the decay constant, the faster the decay process. Knowing the decay constant allows us to calculate the half-life and predict the behavior of isotopes over time in scientific and industrial applications.

Argon Isotopes

Argon isotopes are variants of the element argon, differing in the number of neutrons. Argon has many isotopes, but Argon-41 is particularly interesting due to its use in industrial and scientific applications.

• Argon-41 is radioactive and is commonly used as a tracer for studying gas flow and environmental monitoring.
• Its properties make it useful for tracking and measuring dynamics in various systems.

Understanding isotopes like Argon-41 is important because it helps scientists and engineers employ them effectively in experiments and practical applications. Having insight into their half-lives and decay constants aids in developing strategies for their safe and effective use in a variety of fields.

Exponential Decay Formula

The exponential decay formula is key to unraveling how quantities of radioisotopes, such as Argon-41, reduce over time. The formula is \[ N_t = N_0 e^{-\lambda t} \]where

• \( N_t \) is the amount remaining after time \( t \).
• \( N_0 \) is the initial amount.
• \( \lambda \) is the decay constant.
• \( t \) is the time elapsed.

To find how long it takes for only \(1\%\) of Argon-41 to be left, set up the equation with \( N_0 = 100 \) and \( N_t = 1 \). By solving \[ 1 = 100 e^{-6.3 \times 10^{-3}t} \] and using logarithms, you can isolate and solve for \( t \). This formula and process allow us to predict decay patterns, which is invaluable in fields such as radiometric dating, nuclear medicine, and environmental science.