Problem 74 - University Physics with Modern Physics

Question

An Oceanographic Tracer. Nuclear weapons tests in the 1950 s and 1960 s released significant amounts of radioactive tritium \((^{3}_{1} \mathrm{H},\) half-life 12.3 years \()\) into the atmosphere. The tritium atoms were quickly bound into water molecules and rained out of the air, most of them ending up in the ocean. For any of this tritium-tagged water that sinks below the surface, the amount of time during which it has been isolated from the surface can be calculated by measuring the ratio of the decay product, \(^{3}_{2} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \(_{2}^{3} \mathrm{He}\) to \(_{1}^{3} \mathrm{H}\) in a sample of water is \(1 : 1,\) the water has been below the surface for one half-life, or approximately 12 years. This method has provided oceanographers with a convenient way to trace the movements of subsurface currents in parts of the ocean. Suppose that in a particular sample of water, the ratio of \(_{2}^{3}\) He to \(_{1}^{3} \mathrm{H}\) is 4.3 to 1.0. How many years ago did this water sink below the surface?

Step-by-Step Solution

Answer

The water sank below the surface approximately 29.6 years ago.

Step 1: Understand the Problem

We have the ratio of tritium's decay product, ^3_2 He, to remaining tritium, ^3_1 H, as 4.3:1. The goal is to determine how many years the water has been isolated from the surface based on this ratio. The problem provides the half-life of tritium as 12.3 years.

Step 2: Set Up the Exponential Decay Equation

The decay of tritium follows the exponential decay formula:\[ N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} \] where \( N(t) \) is the remaining quantity of tritium, \( N_0 \) is the initial quantity, \( t \) is the elapsed time in years, and \( T_{1/2} \) is the half-life. Here, \( T_{1/2} = 12.3 \) years.

Step 3: Express Ratio in Terms of Decay Formula

The current ratio of \( ^3_2 \mathrm{He} \) to \( ^3_1 \mathrm{H} \) is given as 4.3:1. This means that the tritium has decayed to produce 4.3 times its original amount that remains. Thus, \( \frac{N_0 - N(t)}{N(t)} = 4.3 \). Simplifying gives \( N_0 = 5.3N(t) \).

Step 4: Solve for Time Using Ratio

Using \( N_0 = 5.3N(t) \) in the decay formula: \[ 5.3N(t) = N(t) \left(\frac{1}{2}\right)^{t/12.3} \]Dividing both sides by \( N(t) \):\[ 5.3 = \left(\frac{1}{2}\right)^{t/12.3} \] Take the natural logarithm of both sides:\[ \ln(5.3) = \frac{t}{12.3} \ln(0.5) \] Now, solve for \( t \):\[ t = \frac{12.3 \cdot \ln(5.3)}{\ln(0.5)} \]

Step 5: Calculate the Time Elapsed

Compute \( t \) using a calculator:\[ \ln(5.3) \approx 1.6677 \]\[ \ln(0.5) \approx -0.6931 \]\[ t = \frac{12.3 \cdot 1.6677}{-0.6931} \approx 29.6 \] So, the tritium-tagged water has been isolated from the surface for approximately 29.6 years.

Key Concepts

Half-life

When we talk about the concept of half-life, we are essentially referring to the amount of time it takes for a certain quantity of a radioactive substance to reduce to half its initial amount through natural radioactive decay. This concept is crucial in understanding how long a radioactive element like tritium will remain active. In the context of exponential decay, the half-life provides a predictable interval over which the substance loses half of its mass, allowing us to calculate how much of it will remain after multiple half-lives.

The formula for calculating the remaining amount of a substance due to decay is given by:

• \( N(t) = N_0\left(\frac{1}{2}\right)^{t/T_{1/2}} \)

where:

• \( N(t) \) is the remaining quantity after time \( t \).
• \( N_0 \) is the initial quantity.
• \( T_{1/2} \) is the half-life of the substance.

By knowing the half-life of tritium (12.3 years in this example), we can compute how long tritium has been decaying in a substance by analyzing the current ratio of tritium to its decay product. This exercise demonstrates that understanding half-life allows scientists to backtrack and figure out the time elapsed since a radioactive material was last at its maximum concentration.

Radioactive Tritium

Radioactive tritium (\(^{3}_{1} \mathrm{H} \)) is an isotope of hydrogen with a nucleus containing one proton and two neutrons, making it unstable and radioactive. Tritium is naturally produced in the atmosphere when cosmic rays interact with atmospheric gases, but it also saw significant increases in levels due to nuclear weapons testing in the mid-20th century. Tritium has a half-life of 12.3 years, meaning it takes this amount of time for half of any given sample of tritium to decay into\(^{3}_{2} \mathrm{He} \) (helium-3).

Tritium is quite unique in its utility as it can easily combine with oxygen to form tritiated water, which behaves like ordinary water, diffusing rapidly throughout the environment, including oceans. This property makes tritium an excellent tracer for studying oceanographic processes like deep-sea current flows and ventilation rates. By knowing the ratios of tritium to its decay product helium-3 in ocean water, scientists can determine how long the water has been isolated from the surface.

Oceanographic Tracers

Oceanographic tracers are substances used by scientists to trace the movement and properties of water masses in the ocean. Tritium is particularly valuable as a tracer for oceanographic studies due to its radioactive nature and origin from nuclear fallout. When tritium is released into the environment, it spreads through atmospheric processes and eventually enters the ocean, providing a "timestamp" for water masses at the time they were isolated from the atmosphere.

By measuring the concentration of tritium and its decay product, helium-3, oceanographers can gain valuable insights into ocean circulation patterns and the timing of water mass movements. One specific advantage of using tritium as a tracer is its half-life, which is long enough to track changes over several decades but short enough to allow for accurate dating of water masses on time scales relevant to climate processes and biological activities in the ocean.

• They help trace ocean currents and the movement of water masses.
• Provide insights into the rates of ocean mixing and renewal.
• Effective for dating water masses and understanding past ocean conditions.

The use of oceanographic tracers like tritium enhances our understanding of complex marine systems, allowing scientists to develop more accurate models of oceanographic processes and predict future changes in ocean behavior.