Suppose that you start with 1000000 atoms of a particular radioactive isotope. how many half-lives woruld be required to reduce the number of undecayed atoms to fewer than 1000?
See your previous post.
To determine the number of half-lives required to reduce the number of undecayed atoms to fewer than 1000, we can use the formula:
N = N₀ * (1/2)^n
Where:
N = number of undecayed atoms after n half-lives
N₀ = initial number of undecayed atoms at the beginning
n = number of half-lives
In this case, N₀ = 1000000 and we want to find the number of half-lives (n) required to reduce N to less than 1000.
Substituting the values into the formula, we get:
1000 = 1000000 * (1/2)^n
To solve for n, we need to isolate the exponential term:
(1/2)^n = 1000 / 1000000
Simplifying the right-hand side:
(1/2)^n = 0.001
To get rid of the exponent, we can take the logarithm of both sides. Let's use the logarithm base 2 since we have a fraction of 1/2 raised to a power:
log₂((1/2)^n) = log₂(0.001)
Using the logarithmic property log₂(x^a) = a * log₂(x):
n * log₂(1/2) = log₂(0.001)
The logarithm of 1/2 to the base 2 is -1, so the equation becomes:
n * (-1) = log₂(0.001)
Multiplying both sides by -1:
n = -log₂(0.001)
Now, we can calculate the value of n:
n ≈ -(-9.966) (using logarithmic calculator)
n ≈ 9.966
Therefore, we need approximately 10 half-lives (rounded up) to reduce the number of undecayed atoms to fewer than 1000.