Question

Cesium- 137 is a radioactive isotope released as a result of the Fukushima Daiichi nuclear disaster in Japan in 2011 . If \(89.2 \%\) remains after \(5.00\) years, what is the half-life?

Short answer

The half-life of Cesium-137 is approximately 30 years.

Step-by-step solution

Understand Decay Formula

Radioactive decay can be described using the formula \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the remaining quantity after time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant. We know \( N(t)/N_0 = 0.892 \) after 5 years.

Calculate Decay Constant

Using the equation \( 0.892 = e^{-k imes 5} \), take the natural logarithm of both sides to solve for \( k \): \[ k = -\frac{\ln(0.892)}{5} \approx 0.0231 \text{ per year} \].

Use Decay Constant to Find Half-Life

The half-life \( T_{1/2} \) is the time it takes for half of the isotope to decay, calculated using the formula \( T_{1/2} = \frac{\ln(2)}{k} \). Substitute \( k = 0.0231 \): \[ T_{1/2} \approx \frac{0.693}{0.0231} \approx 30 \] years.

Validate Calculation

Revisit calculations to ensure accuracy: 1. Calculated decay constant \( k = 0.0231 \). 2. Calculated half-life \( T_{1/2} \approx 30 \text{ years} \). These steps confirm that the solution works mathematically.

Key concepts

Radioactive decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process occurs spontaneously and results in the transformation of an unstable isotope into a more stable form.
Radioactive decay is probabilistic in nature, meaning it's impossible to predict when a particular atom will decay.
However, it is possible to predict the decay behavior of a large number of atoms over time.

• The decay process causes the atom to emit either alpha particles, beta particles, or gamma rays.
• This is essential for understanding how radionuclides behave in the environment.
• Over time, the amount of radioactive material decreases, leading to a reduction in radioactivity.

Understanding radioactive decay is crucial in many fields including nuclear medicine, where radioactive isotopes are used for diagnosis and treatment, and in environmental science to understand the fate of pollutants.

Decay formula

The decay formula is used to model the process of radioactive decay mathematically. It helps in predicting how much of a radioactive substance will remain after a certain amount of time.
The decay formula is represented as: \[ N(t) = N_0 e^{-kt} \] where:

• \(N(t)\) is the remaining quantity of the substance at time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(e\) is the base of the natural logarithm.
• \(k\) is the decay constant, which is unique to each substance and determines how quickly it decays.

For the decay constant \(k\), it can be calculated via the formula:\[ k = -\frac{\ln(N(t)/N_0)}{t} \]Once \(k\) is known, the half-life \(T_{1/2}\) can be found using:\[ T_{1/2} = \frac{\ln(2)}{k} \]This allows us to determine the time required for the substance to reduce to half its original amount. Understanding these formulas is essential for calculating the decay rates and half-lives of radioactive materials.

Fukushima nuclear disaster

The Fukushima nuclear disaster, occurring in March 2011, was one of the most severe nuclear incidents in history. Triggered by a massive earthquake and subsequent tsunami, this disaster led to significant releases of radioactive materials into the environment.
Among the isotopes released was Cesium-137, a radioactive nuclide with a half-life of approximately 30 years.

• Cesium-137 is of particular concern due to its long half-life and its ability to spread across a wide area.
• It can persist in the environment, affecting ecosystems and human health for decades.

The disaster highlighted the need for improved safety protocols in nuclear facilities and led to a re-evaluation of nuclear energy policies worldwide. Understanding the half-life of released isotopes like Cesium-137 is crucial for assessing the long-term impact on the environment and human health, allowing scientists and policymakers to develop strategies for mitigation and recovery.