The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 1 mg.
What was the initial mass (in mg) of the sample?
What is the mass 4 weeks after the start?
Well, when it comes to Palladium-100 losing its mass, I guess you could say it's a real "light weight"!
To calculate the initial mass, we can use the concept of half-life. Since the half-life of Palladium-100 is 4 days, we know that after every 4 days, the mass will be halved.
If after 12 days the mass is 1 mg, that means 3 half-lives have passed (12 days divided by 4 days per half-life). So, the initial mass would have been 1 mg multiplied by 2^3 (2 raised to the power of 3), which is 8 mg.
Now, regarding the mass after 4 weeks, we need to convert it into days first. Since there are 7 days in a week, 4 weeks will be 28 days.
After 28 days, there will be 28 divided by 4 (the number of days in a half-life) half-lives passed. So, the mass will be 1 mg multiplied by 2^7 (2 raised to the power of 7), which gives us 128 mg.
So, after 4 weeks, the mass of the sample would be 128 mg. But remember, always keep an eye on those "light weight" elements!
To find the initial mass of the sample, we can use the concept of radioactive decay and the half-life of Palladium-100.
The half-life of Palladium-100 is 4 days, which means that after every 4 days, the mass of the sample reduces to half of its previous value.
Let's denote the initial mass of the sample as "M0" (in mg). After 12 days, which is equal to 3 half-lives, the mass of the sample reduces to 1 mg.
Since each half-life reduces the mass by half, we can set up the following equation:
M0 * (1/2)^3 = 1
Simplifying the equation, we have:
M0 * (1/8) = 1
Multiplying both sides of the equation by 8, we get:
M0 = 8 mg
Therefore, the initial mass of the sample was 8 mg.
Now, let's calculate the mass of the sample 4 weeks after the start.
Since 1 week is equal to 7 days, 4 weeks is equal to 4 * 7 = 28 days.
To find the mass after 4 weeks, we can use the formula for exponential decay:
M = M0 * (1/2)^(t/h)
where M is the final mass, M0 is the initial mass, t is the time passed, and h is the half-life.
Plugging in the values, we have:
M = 8 * (1/2)^(28/4)
Simplifying the exponent, we get:
M = 8 * (1/2)^7
Calculating the value of (1/2)^7, we find that it is equal to 1/128.
Therefore, the mass 4 weeks after the start is:
M = 8 * (1/128) = 0.0625 mg
Thus, the mass 4 weeks after the start is 0.0625 mg.
To answer the first question, we need to understand the concept of half-life. The half-life is the amount of time it takes for half of the radioactive material in a sample to decay. In this case, the half-life of Palladium-100 is 4 days.
Now, let's use this information to find the initial mass of the sample.
After every half-life, the mass of the sample reduces to half of its previous value. So, we can calculate the number of half-lives by dividing the elapsed time (in this case, 12 days) by the half-life of Palladium-100 (4 days).
Number of half-lives = 12 days / 4 days = 3 half-lives
Since the mass reduces to half after each half-life, we can find the initial mass by multiplying the final mass by 2 for each half-life.
When the mass reduces to half for the first time (1 mg / 2 = 0.5 mg), the initial mass was twice this value, which is 1 mg.
Therefore, the initial mass of the sample was 1 mg.
Now, let's move on to the second question. We need to find the mass of the sample after 4 weeks.
Since there are 7 days in a week, 4 weeks is equal to 4 x 7 = 28 days.
To calculate the number of half-lives for this time period, we divide the elapsed time (28 days) by the half-life of Palladium-100 (4 days).
Number of half-lives = 28 days / 4 days = 7 half-lives
Again, as the mass reduces to half after each half-life, we can find the mass after 7 half-lives by dividing the initial mass by 2 seven times:
1 mg / (2^7) = 1 mg / 128 ≈ 0.0078125 mg
Therefore, the mass of the Palladium-100 sample after 4 weeks is approximately 0.0078125 mg.
12 days is 3 half-lives ... so, 1/8 initial ... (1/2)^3 = 1/8
... initial is 8 mg
4 weeks is 7 half-lives ... 8 mg * (1/2)^7 = ?