If 250 mg of a radioactive element decays to 200 mg in 48 hours, find the half-life of the element
amountleft=originalamount*e^-.692t/thalf
200=250 e^-.692*48/thalf
200/250=e^( )
take the ln of each side
-.2231=-.692*48/thalf
solve for half life time.
To find the half-life of an element, we need to understand the concept of radioactive decay.
Radioactive decay is a process in which unstable atomic nuclei break down and release radiation in the form of particles or electromagnetic waves. The rate at which a radioactive substance decays can be described by its half-life.
The half-life of a radioactive element is the time it takes for half of the initial amount of the substance to decay. In this case, we know that 250 mg of the radioactive element decayed to 200 mg in 48 hours.
To find the half-life, we can set up an equation:
Original amount * (1/2)^(number of half-lives) = Final amount
Let's solve for the number of half-lives it takes for the original amount to decay to the final amount:
250 mg * (1/2)^(number of half-lives) = 200 mg
Dividing both sides of the equation by 250 mg, we get:
(1/2)^(number of half-lives) = 200 mg / 250 mg
Simplifying, we have:
(1/2)^(number of half-lives) = 4/5
To isolate the exponent, we need to take the logarithm of both sides of the equation. Since we are interested in finding the number of half-lives, we can use the logarithm base 1/2 (the base of the exponential term):
log base 1/2 (1/2)^(number of half-lives) = log base 1/2 (4/5)
On the left side, the logarithm base and the exponential base cancel out, resulting in:
number of half-lives = log base 1/2 (4/5)
Using the change of base formula, we can rewrite the equation as:
number of half-lives = log base 10 (4/5) / log base 10 (1/2)
Calculating the logarithms using a calculator, we find:
number of half-lives ≈ 0.321928 / (-0.30103)
number of half-lives ≈ -1.068
Since the number of half-lives cannot be negative, we discard this result and consider the absolute value, giving us:
number of half-lives ≈ 1.068
Therefore, it takes approximately 1.068 half-lives for the original amount of the radioactive element to decay to 200 mg.
To find the half-life, we multiply the number of half-lives by the time it took for the decay to occur (48 hours in this case):
half-life ≈ 1.068 * 48 hours
half-life ≈ 51.264 hours
Therefore, the half-life of the radioactive element is approximately 51.264 hours.