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The decay constant, often represented by the Greek letter \( \lambda \), is a fundamental concept within the study of radioactive decay. This constant is specific to each radioactive material and determines how quickly the substance decays over time.

• A higher decay constant means the substance will decay faster.
• A smaller decay constant indicates a slower rate of decay.

For example, in our exercise, radioactive element A has a decay constant \( \lambda \), while element B has a tenfold larger decay constant, \( 10\lambda \). This tells us that element B will decay much faster than element A. It is vital to understand that the decay constant is intimately linked to the half-life of a radioactive material, which is the time taken for half of the original amount of the substance to decay. The greater \( \lambda \), the shorter the half-life. Understanding the concept of the decay constant enables us to predict and calculate the behavior of substances over time.

The exponential decay formula is a key mathematical representation used in determining how the number of radioactive atoms decreases with time. It is expressed as \[ N(t) = N_0 e^{-\lambda t} \] where:

• \(N(t)\) is the number of atoms remaining after time \(t\).
• \(N_0\) is the initial number of atoms.
• \(\lambda\) is the decay constant.
• \(t\) is the time elapsed.

This formula signifies that as time progresses, the number of atoms reduces exponentially. In the given problem, we're dealing with two radioactive elements (A and B), both having distinct decay constants. By substituting the respective decay constants and the time \(\frac{1}{9\lambda}\) into the formula, students can calculate the remaining atoms of each element. It's fascinating how this formula provides insight into the natural decay process, giving us a clear picture of how rapidly or slowly a substance breaks down over time.

The concept of finding the ratio of radioactive atoms is essential in comparing how different substances decay relative to each other over time. In exercises like ours, after calculating the remaining number of atoms of each element using the exponential decay formula, we can determine their ratio by dividing the number of atoms of one element by another. In the exercise, after time \(\frac{1}{9\lambda}\), we find:

• The number of atoms remaining for element A is \( n e^{-\frac{1}{9}} \).
• The number of atoms remaining for element B is \( n e^{-\frac{10}{9}} \).

To find the ratio of atoms of A to B:\[\frac{N_A}{N_B} = \frac{n e^{-\frac{1}{9}}}{n e^{-\frac{10}{9}}} = e^{-\frac{1}{9} + \frac{10}{9}} = e^{1} = e\]This calculation reveals that even with different decay constants and times, the remaining atoms can be compared easily using their ratios. The output, in this case, gives us clarity in understanding the relative abundance of each element's atoms after a specific period.