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An exponential model is a type of mathematical representation that showcases how quantities change over time. In the context of radioactive decay, substances decay exponentially, meaning their quantity decreases rapidly at first and then more slowly over time. The general form of the exponential decay model is given by the equation

• \( M(t) = M_0 \cdot e^{-kt} \)

where:

• \( M(t) \) is the amount of substance remaining at time \( t \),
• \( M_0 \) is the initial amount of the substance, and
• \( k \) is the decay constant, which characterizes the rate of decay.

This model applies universally across different radioactive materials. The key feature is the exponent \(-kt\), which leads to a continuous and smooth decline of the substance over time. Understanding the exponential nature ensures accuracy in predicting the remaining mass of a sample.

The half-life of a radioactive substance is the time required for half of the initial quantity to decay. It's an intrinsic property of the substance and does not change, regardless of the initial amount. In this exercise, the half-life of radium-226 is given as 1600 years.

• This means that every 1600 years, the amount of radium-226 will reduce to half its previous amount.

The relationship between half-life and decay constant \( k \) is expressed by the equation:

• \( k = \frac{\ln(2)}{\text{half-life}} \)

This crucial ratio allows us to calculate the decay constant for specific substances when their half-life is known, facilitating the use of the exponential model.

The decay constant \( k \) is a crucial parameter in the exponential decay formula. It describes how quickly a substance undergoes radioactive decay. A higher decay constant indicates a faster decay process. The relationship between \( k \) and the half-life \( \tau \) of a substance is given by:

• \( k = \frac{\ln(2)}{\tau} \)

For radium-226, with a half-life of 1600 years, the decay constant is calculated as:

• \( k \approx 0.0004331 \)

Incorporating this decay constant into the exponential model helps ascertain the remaining mass \( M(t) \) at any given time. It is a pivotal factor in projecting both short-term and long-term decreases in radioactive material quantities.

The mass function is an equation providing the remaining mass of a radioactive substance after a period of time \( t \). Using the exponential decay model, we substitute the initial mass and decay constant to form the function:

• \( M(t) = M_0 \cdot e^{-kt} \)

For the given problem with radium-226, the initial mass \( M_0 = 22 \) mg and the decay constant \( k = 0.0004331 \), hence the mass function is:

• \( M(t) = 22 \cdot e^{-0.0004331t} \)

This function is our tool to calculate the remaining mass over time. For example, after 4000 years, substituting \( t = 4000 \) provides \( M(4000) \approx 7.1 \) mg. Using this function, we can also determine when only a certain mass, such as 18 mg, will remain, by solving for \( t \) given \( M(t) = 18 \). This versatility makes the mass function a crucial element in understanding radioactive decay.