Problem 37 - Chemistry: An Atoms-Focused Approach

Question

Fukushima Disaster Explosions at a disabled nuclear Sower station in Fukushima, Japan, in 2011 may have released more cesium- 137 \(\left(t_{1 / 2}=30.2 \text { years }\right)\).Into the ocean than any other single event. How long will it take the radioactivity of this radionuclide to decay to \(5.0 \%\) of the level released in \(2011 ?\)

Step-by-Step Solution

Answer

Answer: It will take approximately 152.5 years for the radioactivity of cesium-137 to decay to 5.0% of its initial level.

Step 1: Understand the radioactive decay formula

The radioactive decay formula is given by: \(N(t)=N_0 e^{-\lambda t}\), where \(N(t)\) is the amount of the substance remaining at time \(t\), \(N_0\) is the initial amount of the substance, \(\lambda\) is the decay constant, and \(t\) is the time elapsed.

Step 2: Find the decay constant using the half-life

The decay constant \(\lambda\) can be found by using the given half-life \(t_{1/2}\): \(\lambda = \frac{\ln 2}{t_{1/2}}\) Substitute the given half-life \(t_{1/2} = 30.2\) years to find the decay constant: \(\lambda = \frac{\ln 2}{30.2} = 0.023\)

Step 3: Determine the desired final amount as a fraction of the initial amount

We are asked to find the time it takes for the radionuclide to decay to 5.0% of the initial amount. Mathematically, this can be written as: \(N(t) = 0.05 N_0\)

Step 4: Substitute the decay constant and desired final amount into the decay formula and solve for time

Using the decay formula and substituting the decay constant and desired final amount, we get: \(0.05 N_0 = N_0 e^{-0.023 t}\) Since we are only interested in the time, we can divide both sides by \(N_0\): \(0.05 = e^{-0.023 t}\) Now, solve for \(t\) by taking the natural logarithm of both sides: \(\ln 0.05 = -0.023 t\) Finally, divide by \(-0.023\) to find the time required: \(t = \frac{\ln 0.05}{-0.023} \approx 152.5 \text{ years}\)

Step 5: Interpret the result

It will take approximately 152.5 years for the radioactivity of cesium-137 released in the Fukushima disaster to decay to 5.0% of its initial level.

Key Concepts

Decay Constant

In the realm of radioactive decay, understanding the decay constant is crucial. This mystical-sounding term actually represents a fundamental characteristic of a radioactive substance. The decay constant, often denoted by the Greek letter \(\lambda\), is a probability rate. It quantifies how quickly a particular radionuclide decays over time.
The decay constant is a parameter that's influenced by the intrinsic properties of the radioactive material. To put it simply, a higher decay constant means the substance decays faster, while a lower decay constant implies a slower decay process.

• The decay constant is key in predicting the behavior of radioactive substances.
• It is used in various formulas to determine how much of a substance remains after a period of time.

Discovering the decay constant generally involves the use of the half-life, another important concept in radioactive decay.

Half-life

The half-life of a radioactive substance is an intuitive concept. It is the time required for half of the radioactive atoms in a sample to decay. This measure is particularly useful because it provides a consistent way to compare how quickly different substances decay.
In the exercise about the Fukushima disaster, Cesium-137 has a half-life of 30.2 years. This means every 30.2 years, half of any given amount of Cesium-137 will have decayed into another substance.

• Half-life is independent of the initial quantity of the substance.
• It is used to calculate the decay constant using the formula: \(\lambda = \frac{\ln 2}{t_{1/2}}\).

Understanding half-life helps in determining how long it takes for a substance to reduce to a certain level, crucial in fields such as archaeology, medicine, and environmental science.

Exponential Decay Formula

The exponential decay formula is a mathematical expression that describes the reduction of a quantity over time. It is particularly relevant in the context of radioactive decay, where it models the rate at which a substance transforms into another form.
The exercise utilizes the formula \(N(t) = N_0 e^{-\lambda t}\) to calculate how long it will take for the radioactivity of cesium-137 to diminish to 5% of its original level.

• \(N(t)\) is the remaining quantity of the substance at time \(t\).
• \(N_0\) represents the initial quantity of the substance.
• The decay constant \(\lambda\) and time \(t\) dictate the decay process.

The beauty of the exponential decay formula lies in its ability to simplify complex decay processes into understandable calculations. It helps predict future states based on present conditions, which is incredibly useful in planning and analysis across various scientific disciplines.