How many half lives will it take for a sample of a radioactive isotope containing 20000 radioactive nuclei to decay to 2000 radioactive nuclei
Dont cheat
to 1/10?
1/10=e^-.692t
take the ln of each side
ln .1=-.692t
t= 3.327 halflives
To determine the number of half-lives required for a sample to decay from 20000 radioactive nuclei to 2000 radioactive nuclei, we need to apply the concept of radioactive decay and the formula for calculating the number of remaining radioactive nuclei after a given number of half-lives.
The relationship between the number of half-lives (n) and the remaining radioactive nuclei (N) can be expressed as follows:
N = N₀ * (1/2)^n
where:
N₀ is the initial number of radioactive nuclei,
N is the remaining number of radioactive nuclei,
n is the number of half-lives.
In the given case, N₀ = 20000 (initial number of radioactive nuclei) and N = 2000 (desired remaining number of radioactive nuclei).
Let's substitute these values into the equation:
2000 = 20000 * (1/2)^n
Next, we'll solve for the number of half-lives (n):
(1/2)^n = 2000 / 20000
(1/2)^n = 1/10
To remove the exponent, we can take the logarithm (base 1/2) of both sides:
log base (1/2) (1/2)^n = log base (1/2) (1/10)
To simplify, note that the logarithm of a number to the power of n is equal to n times the logarithm of the number:
n * log base (1/2) (1/2) = log base (1/2) (1/10)
The logarithm of any number with the base equal to that number is 1, so we can simplify the equation to:
n * 1 = log base (1/2) (1/10)
n = log base (1/2) (1/10)
To solve this, we can use a logarithm calculator or a scientific calculator with a logarithmic function. In this case, we'll use the logarithmic function on a scientific calculator.
Enter log(1/10) and press the base change key, which is usually labeled "log" or "ln," to bring up the base change option.
log(1/10) base (1/2)
The result will be the number of half-lives:
n ≈ 3.3219
Therefore, approximately 3.3219 half-lives are required for a sample of the radioactive isotope to decay from 20000 to 2000 radioactive nuclei. Since we can't have a fraction of a half-life, we'll round this to the nearest whole number.
Hence, it will take approximately 3 half-lives for the sample to decay from 20000 to 2000 radioactive nuclei.