The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 4 mg. What was the initial mass (in mg) of the sample? What is the mass 5 weeks after the start?
amountremaining=Initialamount*e^(.692t/4)
put in given: amount remaining, 12 days, solve for initial amount.
wgtwgt
To find the initial mass of the sample, we can use the formula for exponential decay:
mass = initial mass * (1/2)^(time/half-life)
Given that the half-life of Palladium-100 is 4 days and the mass after 12 days is 4 mg, we can write:
4 mg = initial mass * (1/2)^(12/4)
Simplifying the equation:
4 mg = initial mass * (1/2)^3
4 mg = initial mass * 1/8
Therefore, dividing both sides by 1/8:
initial mass = 32 mg
So, the initial mass of the sample is 32 mg.
To find the mass 5 weeks (35 days) after the start, we can use the same formula:
mass = initial mass * (1/2)^(time/half-life)
Let's calculate it:
mass = 32 mg * (1/2)^(35/4)
Using a calculator:
mass ≈ 32 mg * (1/2)^8.75
mass ≈ 32 mg * 0.066
Therefore,
mass ≈ 2.11 mg
So, the mass 5 weeks after the start is approximately 2.11 mg.
To answer the first question about the initial mass of the sample, we can use the concept of half-life and exponential decay.
The half-life of a substance is the amount of time it takes for half of the substance to decay. In this case, the half-life of Palladium-100 is given as 4 days. This means that after 4 days, half of the initial amount of Palladium-100 will decay and the remaining half will remain.
Now, let's break down the problem step by step:
Step 1: Calculate the number of half-lives that have passed in 12 days.
Since the half-life of Palladium-100 is 4 days, the number of half-lives that have passed in 12 days is 12/4 = 3.
Step 2: Calculate the fractional amount remaining after 3 half-lives.
After each half-life, the remaining amount is halved. So, after 3 half-lives, the fractional amount remaining is (1/2)^3 = 1/8.
Step 3: Calculate the initial mass of the sample.
We know that the initial mass is reduced to 4 mg after 12 days, and the remaining fraction is 1/8. Therefore, we can set up the equation:
(initial mass) * (1/8) = 4 mg
Solving for the initial mass, we get:
(initial mass) = 4 mg * 8 = 32 mg
So, the initial mass of the sample was 32 mg.
Now let's move on to the second question and calculate the mass 5 weeks after the start.
Step 4: Calculate the number of half-lives that have passed in 5 weeks.
Since there are 7 days in a week, 5 weeks is equal to 5 * 7 = 35 days.
The half-life of Palladium-100 is still 4 days, so the number of half-lives that have passed in 35 days is 35/4 = 8.75.
Step 5: Calculate the fractional amount remaining after 8.75 half-lives.
Again, after each half-life, the remaining amount is halved. Therefore, after 8.75 half-lives, the fractional amount remaining is (1/2)^8.75, which is a fractional value between 0 and 1.
Step 6: Calculate the mass 5 weeks after the start.
We know that the initial mass was 32 mg. Therefore, we can set up the equation:
(initial mass) * ((1/2)^8.75) = unknown mass
Solving for the unknown mass, we can multiply the initial mass by ((1/2)^8.75).
Please note that calculating the exact value of ((1/2)^8.75) involves using logarithms or more advanced techniques.