Iodine-131 has a half-life of 8 days. How many days would it take for a sample of Iodine -131 to go from 50 grams to 3.125 grams?
Note that 3.125/50 = 1/16 = (1/2)^4
That means that 4 half-lives have passed: 32 days
To determine how many days it would take for a sample of Iodine-131 to decay from 50 grams to 3.125 grams, we can use the concept of half-life.
1. Calculate the number of half-lives:
Since the half-life of Iodine-131 is 8 days, we need to determine how many times 8 days will go into the total time it takes for the sample to decay.
Number of half-lives = log(base 2)(initial mass / final mass)
Initial mass = 50 grams
Final mass = 3.125 grams
Number of half-lives = log(base 2)(50 / 3.125)
2. Calculate the total time:
Now, multiply the number of half-lives by the half-life period (8 days) to find the total time it takes for the sample to decay.
Total time = Number of half-lives × half-life period
Let's calculate the number of half-lives first:
Number of half-lives = log(base 2)(50 / 3.125)
Number of half-lives = log(base 2)(16)
Number of half-lives ≈ 4 (approximately)
Now, calculate the total time:
Total time = Number of half-lives × half-life period
Total time = 4 × 8
Total time = 32 days
Therefore, it would take approximately 32 days for a sample of Iodine-131 to decay from 50 grams to 3.125 grams.
To find the number of days it would take for a sample of Iodine-131 to go from 50 grams to 3.125 grams, we can use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / T)
Where:
N(t) is the amount of the substance at time t
N₀ is the initial amount of the substance
t is the time elapsed
T is the half-life of the substance
In this case, N₀ is 50 grams, N(t) is 3.125 grams, and T is 8 days. Let's rearrange the formula to solve for t:
N(t) = N₀ * (1/2)^(t / T)
3.125 = 50 * (1/2)^(t / 8)
To solve for t, we need to isolate the exponent. We can start by taking the logarithm (base 1/2) of both sides of the equation:
log base 1/2 (3.125) = log base 1/2 (50 * (1/2)^(t / 8))
Since log base 1/2 (1/2) equals 1, we can simplify the right side:
log base 1/2 (3.125) = log base 1/2 (50) + log base 1/2 ((1/2)^(t / 8))
Next, we simplify the right side further by using the property of logarithms that states logₐ(b^c) = c * logₐ(b):
log base 1/2 (3.125) = log base 1/2 (50) + (t / 8) * log base 1/2 (1/2)
Since log base 1/2 (1/2) equals 1, the equation becomes:
log base 1/2 (3.125) = log base 1/2 (50) + (t / 8)
Now, we can subtract log base 1/2 (50) from both sides:
log base 1/2 (3.125) - log base 1/2 (50) = t / 8
By simplifying the left side of the equation:
log base 1/2 (3.125 / 50) = t / 8
Using logarithm properties, we convert the equation to exponential form:
1/2^(t / 8) = 3.125 / 50
Solving for t, we can multiply both sides by 8 and then take the logarithm (base 1/2) of both sides:
log base 1/2 (1/2^(t / 8)) = log base 1/2 (3.125 / 50)
Simplifying the left side and evaluating the right side:
t / 8 = log base 1/2 (0.0625)
Finally, we solve for t by multiplying both sides by 8:
t = 8 * log base 1/2 (0.0625)
Using a logarithm calculator or software, we find that log base 1/2 (0.0625) is approximately equal to 6. Therefore:
t ≈ 8 * 6
t ≈ 48
Therefore, it would take approximately 48 days for a sample of Iodine-131 to go from 50 grams to 3.125 grams.