Problem 55 - Chemistry Matter and Change

Question

Nuclear mishap The half-life of tritium \(\left(_{1}^{3} \mathrm{H}\right)\) is 12.3 \(\mathrm{y}\) If 48.0 \(\mathrm{mg}\) of tritium is released from a nuclear power plant during the course of a mishap, what mass of the nuclide will remain after 49.2 \(\mathrm{y} ?\) After 98.4 \(\mathrm{y} ?\)

Step-by-Step Solution

Answer

After 49.2 years, there will be 3.0 mg of tritium remaining, and after 98.4 years, there will be approximately 0.1875 mg of tritium remaining.

Step 1: Write down the decay equation

The decay equation relates the final mass of a radioactive substance (Mf) to its initial mass (Mi), half-life (t_1/2), and the time elapsed (t): \[Mf = Mi \times (1/2)^{(t / t_{1/2})}\]

Step 2: Calculate the remaining mass after 49.2 years

To find the remaining mass of tritium after 49.2 years, we can plug the given values into the decay equation: Mi = 48.0 mg, t = 49.2 years, and t_1/2 = 12.3 years. \[Mf = 48.0\,\text{mg}\, \times (1 / 2)^{(49.2 / 12.3)}\] Now, solve for Mf: \[Mf = 48.0\, \text{mg} \times (1/2)^{4}\] \[Mf = 48.0\, \text{mg} \times (1/16)\] \[Mf = 3.0\, \text{mg}\] After 49.2 years, there will be 3.0 mg of tritium remaining.

Step 3: Calculate the remaining mass after 98.4 years

Similarly, we can find the remaining mass after 98.4 years: Mi = 48.0 mg, t = 98.4 years, and t_1/2 = 12.3 years. \[Mf = 48.0\,\text{mg}\, \times (1 / 2)^{(98.4 / 12.3)}\] Now, solve for Mf: \[Mf = 48.0\, \text{mg} \times (1/2)^{8}\] \[Mf = 48.0\, \text{mg} \times (1/256)\] \[Mf = 0.1875\, \text{mg}\] After 98.4 years, there will be approximately 0.1875 mg of tritium remaining.

Key Concepts

Half-Life

Understanding the concept of half-life is essential when studying nuclear chemistry and radioactive decay. It refers to the time required for half of the radioactive atoms in a sample to undergo decay. This decay rate is constant for a given isotope, meaning that no matter how much of the substance you have, half of it will decay over one half-life period.

To visualize, let's consider a 100 mg sample of a substance with a half-life of 5 years. After the first 5 years, only 50 mg would remain. After another 5 years (a total of 10 years), you would have 25 mg left, and so on. This inherent predictability makes it possible to calculate the amount of radioactive material remaining after any number of half-lives.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and processes involving the nucleus of an atom, where isotopes, radioactivity, and nuclear reactions are fundamental topics. Isotopes of an element have the same number of protons but vary in the number of neutrons, affecting their stability.

Radioactive isotopes, such as tritium ( \( _{1}^{3}H \)), used in our exercise, are unstable and emit radiation as they transform into more stable forms. The study of nuclear chemistry not only helps in understanding and calculating the decay of these isotopes over time, as shown in the tritium example, but also has practical applications in energy production, medical diagnostics, and treatments.

Exponential Decay

The mathematical model that describes the process of radioactive decay is known as exponential decay. It's a type of decay where the quantity decreases at a rate proportional to its current value, expressed by the formula \[Mf = Mi \times (1/2)^{(t / t_{1/2})}\], which shows how the mass ( \(Mf\) ) of a radioactive substance decreases over time.

The section above in the problem solution exemplifies how to use the formula to calculate the remaining mass after certain time intervals. Because of the base \( (1/2) \), each time period you add (one half-life), you divide the remaining amount by two, demonstrating the 'halving' property that is characteristic of exponential decay. This concept is pivotal across various scientific disciplines, including physics, biology, and finance, where it describes not only radioactive decay but also population decline, temperatures cooling, and bank interest, respectively.