a certain radioactive isotope has a half life or 850 years.how many years would it take for a given amount of this isotope to decay to 70% of that amount?
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To determine the number of years it would take for a given amount of a radioactive isotope to decay to 70% of that amount, you can use the concept of half-life.
1. Start by calculating the number of half-lives required to decay to 70% of the original amount. Since each half-life reduces the amount by half, we need to find how many times we need to divide the original amount by 2 to reach 70%.
2. The formula to calculate the number of half-lives required is:
Number of half-lives = log(0.7) / log(0.5)
Here, log denotes the logarithm.
3. Calculate the number of half-lives using the formula:
Number of half-lives = log(0.7) / log(0.5) ≈ 1.512
4. Multiply the number of half-lives by the half-life of the isotope to get the total time it would take for decay.
Time required = Number of half-lives * Half-life of the isotope = 1.512 * 850 years ≈ 1281.2 years.
Therefore, it would take approximately 1281.2 years for the given amount of the isotope to decay to 70% of that amount.
To find out how many years it would take for a given amount of a radioactive isotope to decay to 70% of that amount, you can use the formula for exponential decay. The formula is in the form:
N = N₀ * (1/2)^(t / T₁/₂)
Where:
- N is the remaining amount of the isotope after time t
- N₀ is the initial amount of the isotope
- t is the time passed
- T₁/₂ is the half-life of the isotope
Let's express the problem mathematically using this formula:
Since the half-life of the isotope is 850 years, we can substitute T₁/₂ = 850 into the formula. We want to find out how many years it takes for the isotope to decay to 70% of the initial amount, so we let N = 0.7N₀:
0.7N₀ = N₀ * (1/2)^(t / 850)
Now we can solve for t:
0.7 = (1/2)^(t / 850)
To isolate the exponent, we can take logarithms of the equation. Let's use logarithm base 2:
log₂(0.7) = (t / 850) * log₂(1/2)
Using the properties of logarithms, we can rewrite the equation as:
log₂(0.7) = (-t / 850) * log₂(2)
Simplifying the equation:
t = -850 * log₂(0.7) / log₂(2)
Using a calculator, we can evaluate this expression:
t ≈ -850 * (log(0.7) / log(2))
Note that the negative sign in front of 850 just indicates that it will take a negative amount of time for the isotope to decay to 70% of the initial amount. In practical terms, we can simply ignore the negative sign and consider the absolute value of time.
Therefore, to find out how many years it would take for the given amount of the isotope to decay to 70% of that amount, you can substitute the value of log₂(0.7) / log₂(2) into the expression and calculate the result.