I have tried for days to figure out how to work this problem. Please help show me how to find the answer.
A particular radioactive isotope decays from 210 mg to 182.7 mg in 15 days.
Find the half life of the isotope.
To find the half-life of a radioactive isotope, you can use the formula:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
N(t) is the amount of the isotope remaining at time t
N₀ is the initial amount of the isotope
t is the elapsed time
t₁/₂ is the half-life of the isotope
In this case, we know the initial amount N₀ = 210 mg and the final amount N(t) = 182.7 mg. We also know that the elapsed time t = 15 days. We want to find the half-life t₁/₂.
Using the formula, we can rewrite it to solve for t₁/₂:
182.7 = 210 * (1/2)^(15 / t₁/₂)
Now we can solve for t₁/₂.
Divide both sides of the equation by 210:
182.7 / 210 = (1/2)^(15 / t₁/₂)
Simplify the left side:
0.87 = (1/2)^(15 / t₁/₂)
Next, take the logarithm of both sides using the base 2 (since we have 1/2 in the equation):
log₂(0.87) = log₂((1/2)^(15 / t₁/₂))
Using the logarithm rule log₂(a^b) = b * log₂(a):
log₂(0.87) = (15 / t₁/₂) * log₂(1/2)
The logarithm log₂(1/2) is equal to -1, so the equation becomes:
log₂(0.87) = -15 / t₁/₂
Now, isolate t₁/₂ by multiplying both sides by -1 and dividing by log₂(0.87):
t₁/₂ = -15 / (log₂(0.87))
Using a calculator, we can evaluate the expression:
t₁/₂ ≈ -15 / (-0.13997) ≈ 107.16 days
The half-life of the isotope is approximately 107.16 days. Remember, the negative sign indicates that the rate of decay is decreasing.
To find the half-life of a radioactive isotope, we need to use the decay equation. The decay of a radioactive isotope can be modeled by the formula:
A = A₀ * (1/2)^(t / t₁/₂)
Where:
- A is the amount of the isotope remaining after time t
- A₀ is the initial amount of the isotope
- t is the time that has passed
- t₁/₂ is the half-life of the isotope
In this case, we are given the initial amount A₀ as 210 mg. We also know that after 15 days, the amount A has decreased to 182.7 mg. We can use these values to solve for the half-life, t₁/₂.
Let's substitute the known values into the equation:
182.7 mg = 210 mg * (1/2)^(15 / t₁/₂)
Before we solve the equation, let's simplify it further:
182.7 / 210 = (1/2)^(15 / t₁/₂)
Now, take the logarithm of both sides of the equation to get rid of the exponent:
log(182.7 / 210) = log((1/2)^(15 / t₁/₂))
Using logarithmic properties, we can simplify further:
log(182.7 / 210) = (15 / t₁/₂) * log(1/2)
Now, we can solve for t₁/₂ by isolating it:
(15 / t₁/₂) * log(1/2) = log(182.7 / 210)
Divide both sides of the equation by log(1/2):
15 / t₁/₂ = log(182.7 / 210) / log(1/2)
Finally, solve for t₁/₂ by isolating it:
t₁/₂ = 15 / (log(182.7 / 210) / log(1/2))
Now, we can use a calculator to evaluate the right side of the equation to find the value of t₁/₂.
By following these steps, you should be able to calculate the half-life of the isotope. Let me know if you need further assistance!