Question

Plutonium- 239 has a decay constant of \(2.88 \times 10^{-5}\) year \(^{-1}\). What percentage of a \({ }^{239}\) Pu sample remains after 1000 years? After 25,000 years? After 100,000 years?

Short answer

97.16% remains after 1000 years, 48.83% after 25,000 years, 5.62% after 100,000 years.

Step-by-step solution

Understanding the Decay Formula

We use the exponential decay formula, which is given by: \[ N(t) = N_0 imes e^{-kt} \]where \( N(t) \) is the remaining quantity after time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( e \) is the base of the natural logarithm.

Calculation for 1000 Years

Substitute the values into the decay formula:\[ N(t) = N_0 imes e^{-(2.88 imes 10^{-5}) imes 1000} \]Simplify this to:\[ N(t) = N_0 imes e^{-0.0288} \]We calculate \( e^{-0.0288} \) to find the percentage remaining: approximately 97.16% of the original \( ^{239}\text{Pu} \) sample remains after 1000 years.

Calculation for 25,000 Years

Substitute the values into the decay formula:\[ N(t) = N_0 imes e^{-(2.88 imes 10^{-5}) imes 25000} \]Simplify this to:\[ N(t) = N_0 imes e^{-0.72} \]We calculate \( e^{-0.72} \) to find the percentage remaining: approximately 48.83% of the original \( ^{239}\text{Pu} \) sample remains after 25,000 years.

Calculation for 100,000 Years

Substitute the values into the decay formula:\[ N(t) = N_0 imes e^{-(2.88 imes 10^{-5}) imes 100000} \]Simplify this to:\[ N(t) = N_0 imes e^{-2.88} \]We calculate \( e^{-2.88} \) to find the percentage remaining: approximately 5.62% of the original \( ^{239}\text{Pu} \) sample remains after 100,000 years.

Key concepts

Decay Constant

The decay constant is a crucial parameter in the study of radioactive decay. It is represented by the symbol \( k \) in the decay formula and refers to the probability per unit time that a nucleus will decay. For Plutonium-239, the decay constant is given as \( 2.88 \times 10^{-5} \) year\( ^{-1} \). This means that in one year, each nucleus of \(^{239}\text{Pu}\) has a \(0.00288\)% chance of decaying.
Understanding the decay constant helps us predict how quickly a radioactive sample will decrease over time. In mathematical terms, the decay constant is used in the exponential decay formula to calculate how much of the sample remains after a certain time period.

• The larger the decay constant, the faster the decay process.
• The smaller the decay constant, the slower the decay and the longer the substance remains radioactive.

Knowing the decay constant allows us to compute other key characteristics like the half-life, which is the time required for half of the radioactive substance to decay.

Exponential Decay

Exponential decay describes how the quantity of a radioactive material decreases over time. The core idea is that the rate of decay of a sample is proportional to the number of undecayed nuclei present. This relationship is captured by the exponential decay formula: \[ N(t) = N_0 \times e^{-kt} \]where:

• \( N(t) \) is the quantity remaining after time \( t \).
• \( N_0 \) is the initial quantity of the substance.
• \( k \) is the decay constant.
• \( e \) is the base of the natural logarithm (approximately 2.71828).

The formula tells us that the quantity reduces at an exponentially decreasing rate. This means that the speed of decay slows down as time progresses, but never truly reaches zero.
When solving problems like the one given, you substitute the appropriate values into the formula to find the remaining percent of the sample after different times. This pattern of exponential decay leads to the calculation of percentages remaining, showing exactly how the remaining fraction diminishes over time.

Half-Life

Half-life is another vital concept when discussing radioactive materials. It is the time required for half of the radioactive atoms in a sample to decay. In other words, during one half-life, the sample's quantity will reduce to 50% of its original amount.
For any given radioactive substance, its half-life is constant. You can calculate the half-life using the decay constant with the formula:\[ T_{1/2} = \frac{\ln(2)}{k} \]where \( T_{1/2} \) is the half-life and \( \ln(2) \) is the natural logarithm of 2 (approximately 0.693).
By knowing the half-life, you can easily predict how long it will take for a sample to reduce to a desired percentage of its original size.

• The concept of half-life is widely used in different areas, from archaeological dating (using carbon-14) to medicine (with radioactive tracers).
• It's a straightforward way to understand the persistence and longevity of radioactive materials.

For Plutonium-239, the half-life is crucial for understanding how it behaves over thousands of years, making it particularly significant for applications relating to nuclear waste management and historical study.