Final answer:
To calculate the time taken for a 320 g sample to decay to 2.5 g, the half-life of the isotope must be known. The number of half-lives that must pass for this decay is calculated using the exponential decay formula, and then this number is multiplied by the half-life to find the total time required.
Step-by-step explanation:
Calculating the Decay Time for a Radioactive Isotope
To determine how long it would take a 320 g sample to decay to 2.5 g, first, we need to identify the half-life of the element or isotope in question. The half-life is the time required for half of the radioactive substance to decay. Using the half-life, we can calculate the number of half-lives that occur to reach from the starting mass to the end mass, using the formula for exponential decay. Here's a step-by-step guide:
- Identify the initial mass (in this case, 320 grams).
- Identify the final mass desired (in this case, 2.5 grams).
- Determine the half-life of the isotope (not provided in this case, but essential for calculations).
- Use the formula for exponential decay, which is the final amount (A) equals the initial amount (Ao) times one-half to the power of the number of half-lives that have passed (n): A = Ao *(1/2)^n.
- Solve for n, the number of half-lives passed, by rearranging the formula: n = log(A/Ao) / log(0.5).
- Multiply n by the half-life duration to find the total time it takes for the decay to reach the desired mass.
This calculation is a standard process in radioactive decay problems and used in scenarios provided such as the decay of niobium-94, Ra-226, and Ac-225 in various examples.