Calculating the Amount of Radioactive Material The graphing calculator can run a program that graphs the relationship between the amount of radioactive material and elapsed time. Given the half-life of the radioactive material and the initial amount of material in grams, you will graph the relationship between the amount of radioactive material and the elapsed time. Then, with the elapsed time, you will trace the graph to calculate the amount of radioactive material. Go to Appendix C. If you are using a TI-83 Plus, you can download the program RADIOACT and run the application as directed. If you are using another calculator, your teacher will provide you with key-strokes and data sets to use. After you have run the program, answer these questions. a. Determine the amount of neptunium-235 left after 2.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. b. Determine the amount of neptunium-235 left after 5.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. c. Determine the amount of uranium-232 left after 100 years, given the half- life of uranium-232 is 69 years and the initial amount was 10.0 g.

Step by step solution

01

Calculation for 2.0 Years (Neptunium-235)

Use the radioactive decay formula. Here, P=8.00g, t=2.0 years, and h=1.08 years. Therefore, \( A = 8.00 * (0.5)^(2.0/1.08) \).

02

Calculation for 5.0 Years (Neptunium-235)

Re-apply the decay formula, this time with t=5.0 years for the same material Neptunium-235: \( A = 8.00 * (0.5)^(5.0/1.08) \).

03

Calculation for 100 Years (Uranium-232)

Finally, apply the decay formula for different radioactive material Uranium-232 with t=100 years, h=69 years, and P=10.0g: \( A = 10.0 * (0.5)^(100/69) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life

Half-life is a term that describes the amount of time it takes for half of the atoms in a sample of a radioactive material to decay. It is a fundamental property of radioactive substances and varies widely among different substances. In our exercise about neptunium-235, the half-life is given as 1.08 years. This means that every 1.08 years, the amount of neptunium-235 will be reduced to half of its initial quantity.

For instance, if we start with 8 grams of neptunium-235, after 1.08 years, only 4 grams would remain. Following another 1.08 years, that amount would again be halved to 2 grams, and so on. The concept of half-life is crucial in calculating the remaining quantity of radioactive material over time. By using the half-life, we can determine how much radioactive material will be left after a certain period using the decay formula, as demonstrated in the problem's solution.

Radioactive Material

Radioactive materials are substances that undergo spontaneous decay as their atomic nuclei release energy in the form of radiation. This decay process transforms the original material into different elements or isotopes. For example, in our exercise, we look at neptunium-235 and uranium-232, both of which are radioactive isotopes that will decay over time.

The rate at which a radioactive material decays is characterized by its half-life. Understanding the properties and behavior of different radioactive materials is essential in a variety of fields, from nuclear power generation to medical imaging and treatments. When handling radioactive materials, safety is paramount due to the potential hazards associated with radiation exposure.

Exponential Decay

Exponential decay describes the process by which the quantity of a radioactive material decreases over time at a rate proportional to its current value. This means that as time passes, the material decays at a rate that is continuously slowing down. The mathematical formula for exponential decay, particularly for radioactive substances, is normally expressed as: \[ A = P \times (0.5)^{(t/h)} \] where 'A' is the amount of substance left after time 't', 'P' is the initial amount, 'h' is the half-life, and '0.5' represents the half-life decay factor because half of the substance decays each half-life.

This formula encapsulates the principle that, regardless of how much substance remains, it will always take the same amount of time (one half-life) for half of it to decay. This predictability allows us to calculate the amount of a radioactive material that will remain after any given time, as seen in the step-by-step solutions provided in the original exercise. Understanding exponential decay is critical for interpreting radioactive decay graphs and predicting the behavior of radioactive isotopes over time.