Suppose that you are given a 100 gram sample of a radioactive substance with a half-life of 32 days. How many grams will be left after 192 days?
192 days is 6 half-lives.
(1/2)^6 = 1/64
1/64 * 100 g = 1.5625 g
Carbon 14 has a half-life of 5,730 years. How long will it take for 400 grams of carbon 14 to decay to the point that only 100 grams are carbon 14.
To find out how many grams of the radioactive substance will be left after 192 days, we can use the formula for exponential decay:
Amount = Initial amount * (1/2)^(time/half-life)
Given information:
Initial amount (A₀) = 100 grams
Half-life (T₁/₂) = 32 days
Time (t) = 192 days
Now, let's calculate the amount of substance left:
Amount = 100 * (1/2)^(192/32)
Amount = 100 * (1/2)^6
Amount = 100 * 1/64
Amount = 1.5625 grams
Therefore, after 192 days, approximately 1.5625 grams of the radioactive substance will be left.
To determine how many grams will be left after 192 days, we need to calculate the number of half-lives that have passed and then calculate the remaining mass.
1. Determine the number of half-lives: The given half-life is 32 days. Divide the total time (192 days) by the half-life to find the number of half-lives: 192 ÷ 32 = 6 half-lives.
2. Calculate the remaining mass after each half-life:
- After the first half-life: 100 grams / 2 = 50 grams.
- After the second half-life: 50 grams / 2 = 25 grams.
- After the third half-life: 25 grams / 2 = 12.5 grams.
- After the fourth half-life: 12.5 grams / 2 = 6.25 grams.
- After the fifth half-life: 6.25 grams / 2 = 3.125 grams.
- After the sixth half-life: 3.125 grams / 2 = 1.5625 grams.
Therefore, after 192 days, approximately 1.5625 grams of the radioactive substance will be left.