Problem 31 - Essential Calculus

Question

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{239} \mathrm{Pu} \quad 24.100 \quad \quad 2.1 \mathrm{~g} $$

Step-by-Step Solution

Answer

The quantity of \({ }^{239} \mathrm{Pu}\) after 1000 years and 10,000 years can be calculated using the formula for radioactive decay with the provided half-life. The actual values are left to be calculated.

Step 1: Understand the Concept of Half-Life

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. In this case, the half-life of Plutonium-239 is given as 24100 years.

Step 2: Calculate the Amount After 1000 Years

We know that the formula to calculate the remaining quantity after a certain time is given by the formula: \[ N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \] Where, \(N\) is the final quantity, \(N_0\) is the initial quantity, \(t\) is the time, and \(T\) is the half-life. Substituting the given values \(N_0 = 2.1g\), \(T = 24100\), and \(t = 1000\), we get: \[ N = 2.1 \times \left(\frac{1}{2}\right)^{\frac{1000}{24100}} \] Calculate this value to get the quantity after 1000 years.

Step 3: Calculate the Amount After 10,000 Years

Using the same formula and substituting \(t = 10000\) we get \[ N = 2.1 \times \left(\frac{1}{2}\right)^{\frac{10000}{24100}} \] Calculate this value to get the quantity after 10,000 years.

Key Concepts

Radioactive Isotope Decay

Radioactive isotope decay is a natural process where unstable isotopes shed particles from their nucleus to become more stable. This gives rise to a variety of forms of radiation, including alpha, beta, and gamma rays. The decay is not instant, but is predictable over time. When an isotope decays, its nucleus changes. This can transform it into a different element or isotope.
Understanding this decay process is crucial for several industries, from medical therapies to nuclear power. Radioactive decay follows a particular decay pattern based on the half-life of the isotope. In our exercise, we are dealing with Plutonium-239. This isotope has a half-life of 24,100 years, meaning it will take this time for half of any given sample to decay into a different element.
Knowing the half-life of an isotope helps predict how it decays over time, which is essential in various scientific and engineering applications. As time progresses, the quantity of the original radioactive isotope decreases exponentially. This characteristic behavior is what we explore when calculating isotope decay over specific time periods.

Mathematical Formulas in Chemistry

Chemical processes often rely on mathematical formulas to predict how substances change. When dealing with radioactive isotopes, formulas help calculate how much of the substance remains after a given time.
For the decay process, we use the formula: \[ N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} \]

• \( N \) is the quantity after time \( t \).
• \( N_0 \) represents the initial quantity of the isotope, here 2.1 grams for Plutonium-239.
• \( T \) is the half-life, in our example, 24,100 years.
• \( t \) is the time elapsed, 1000 or 10,000 years in the problem.

This equation helps model the decay by predicting the fraction of the initial quantity remaining after a given period. The exponent \( \left(\frac{t}{T}\right) \) determines how many half-lives have passed. Thus, these calculations make it easier to understand how much of a radioactive substance will remain.

Exponential Decay Formula

The exponential decay formula is a powerful tool used to describe the reduction of a quantity over time. Specifically, for radioactive decay, this formula shows how quickly an isotope decreases in quantity. In our exercise, this formula is key to determining how much Plutonium-239 remains after specific periods like 1000 and 10,000 years.
The exponential decay formula gives us:\[N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}\]This equation indicates that as \( t \) increases, the exponent \( \left(\frac{t}{T}\right) \) increases, causing the base, \( (1/2) \), to decrease when raised to higher powers.

• This implies smaller fractions of the initial quantity remain over longer periods.
• The exponential nature reflects the continual halving process.

Each time interval \( T \) represents another half-life, showing the decay pattern's systematic and predictable nature. This feature is what makes the exponential decay formula invaluable in fields that require precise decay predictions, including environmental science, archaeology, and medicine.