a sample of a radioisotope had a mass of 100.0 grams. After exactly 24 days, 6.25 grams remained. The half life of the isotope is _________
To find the half-life of the radioisotope, we can use the information given:
1. Start with the initial mass of the radioisotope, which is 100.0 grams.
2. After exactly 24 days, the remaining mass of the radioisotope is 6.25 grams.
Now, let's calculate the number of half-lives:
3. The remaining mass after one half-life can be calculated by dividing the initial mass by 2:
Remaining mass after 1 half-life = 100.0 grams / 2 = 50.0 grams
4. The remaining mass after two half-lives can be obtained by dividing the mass after one half-life by 2:
Remaining mass after 2 half-lives = 50.0 grams / 2 = 25.0 grams
5. Similarly, the remaining mass after three half-lives is:
Remaining mass after 3 half-lives = 25.0 grams / 2 = 12.5 grams
6. Continuing this pattern, we can calculate the remaining mass after four half-lives, which is:
Remaining mass after 4 half-lives = 12.5 grams / 2 = 6.25 grams
Since the remaining mass after four half-lives is equal to the observed mass after 24 days, we can conclude that the half-life of the radioisotope is 24 days.
To find the half-life of the radioisotope, you can use the formula:
t1/2 = (t * log(2)) / log(N0 / Nt)
Where:
- t1/2 is the half-life of the isotope
- t is the elapsed time in the given units (24 days in this case)
- log represents the logarithm function (base 10)
- N0 is the initial quantity or mass of the isotope (100.0 grams)
- Nt is the remaining quantity or mass of the isotope (6.25 grams)
Now, let's substitute the known values into the formula and calculate the half-life:
t1/2 = (24 * log(2)) / log(100.0 / 6.25)
Using a calculator, you can solve this equation:
t1/2 ≈ (24 * 0.3010) / (2.0000 - 0.7960)
t1/2 ≈ 7.23 / 1.2040
t1/2 ≈ 6.00 days
Therefore, the half-life of the radioisotope is approximately 6.00 days.
ln(No/N) = kt
No = 100
N = 6.25
t = 24 days.
Solve for k, then substitute into the equation below.
k = 0.693/t1/2 and solve for t1/2