An unknown radioactive element decays into non-radioactive substances. In 420 days the radioactivity of a sample decreases by 39 percent.
1.What is the half-life of the element?
2.How long will it take for a sample of 100 mg to decay to 46 mg?
A(t)= Ao* (1/2)^(t/h)
where h is the half life.
A/Ao = 1-0.39 = 0.61
0.61 = (1/2)^(420/h)
log 0.61 = (420/h) log (1/2)
420/h = 0.71312
h = 589 days
When 100 mg becomes 46 mg of radioactive material, A/Ao = 0.46
0.46 = (1/2)^(t/589)
t/589 = log(0.46)/log(0.50)= 1.1203
t = 660 days
1. Well, I guess this unknown radioactive element is going through a bit of a depreciation crisis, isn't it? The half-life is the time it takes for half of the radioactive substance to decay. So, if it decreases by 39 percent in 420 days, we can calculate that each time it decreases by half, the time it takes is 420 days divided by 39 percent...but I'm not very good with math, so I'll leave that part up to you!
2. Ah, the decay of the unknown radioactive element continues! Now, if we start with 100 mg and want to know when it will decay to 46 mg, we need to calculate how many half-lives it would take. Each time it decays by half, it's like cutting a slice off a shrinking pie. So, divide the initial amount by the final amount, and that will give you the number of half-lives. Then, multiply this by the half-life time to get your answer. But remember, I'm just here to provide a little lightheartedness, not do math!
To find the half-life of the unknown radioactive element, we can use the formula:
\(N = N_0 × (1/2)^{t/h}\)
Where:
- N is the final amount of the substance
- N0 is the initial amount of the substance
- t is the time passed
- h is the half-life of the substance
We are given that the radioactivity of the sample decreases by 39 percent in 420 days, which means the final amount (N) is 61 percent (100% - 39%) of the initial amount (N0).
Using this information, we can set up the following equation:
\(0.61 = (1/2)^{420/h}\)
To solve for the half-life (h), we can take the logarithm of both sides of the equation base 2:
\(log_2(0.61) = log_2((1/2)^{420/h})\)
\(log_2(0.61) = (420/h) × log_2(1/2)\)
\(log_2(0.61) = (420/h) × (-1)\)
\(log_2(0.61) = -420/h\)
Now, we can isolate the half-life (h) by rearranging the equation:
\(h = - (420 / log_2(0.61))\)
Calculating this expression will give us the value of the half-life of the unknown radioactive element.
To calculate how long it will take for a sample of 100 mg to decay to 46 mg, we can again use the formula mentioned above:
\(N = N_0 × (1/2)^{t/h}\)
In this case, the initial amount (N0) is 100 mg, and the final amount (N) is 46 mg. We already know the value of the half-life (h) from the previous calculation.
Plugging these values into the formula, we can solve for t:
\(46 = 100 × (1/2)^{t/h}\)
Rearranging the equation:
\(0.46 = (1/2)^{t/h}\)
Taking the logarithm of both sides base 2:
\(log_2(0.46) = log_2((1/2)^{t/h})\)
\(log_2(0.46) = (t/h) × log_2(1/2)\)
\(log_2(0.46) = (t/h) × (-1)\)
\(log_2(0.46) = -t/h\)
Now, we can isolate t by rearranging the equation:
\(t = - (log_2(0.46) × h)\)
Calculating this expression will give us the time it takes for the sample to decay from 100 mg to 46 mg.
To determine the half-life of the unknown radioactive element, we need to use the formula for exponential decay:
A = A0 * (1/2)^(t / T)
Where:
- A is the final amount of the substance
- A0 is the initial amount of the substance
- t is the time that has passed
- T is the half-life of the substance
Now, let's solve for T using the information given in the question:
1. We know that the radioactivity decreases by 39 percent in 420 days. This means that A = 0.61 * A0.
Substituting the values into the equation, we get:
0.61 * A0 = A0 * (1/2)^(420 / T)
Dividing both sides by A0, we can eliminate it:
0.61 = (1/2)^(420 / T)
To isolate T, we can take the logarithm of both sides (base 1/2):
log(0.61) = (420 / T) * log(1/2)
Now, we can solve for T by dividing both sides by (420 * log(1/2)):
T = -420 / (log(0.61) / log(1/2))
2. To find out how long it will take for a sample of 100 mg to decay to 46 mg, we can use the same exponential decay formula:
A = A0 * (1/2)^(t / T)
Substituting the values into the equation, we get:
46 = 100 * (1/2)^(t / T)
Dividing both sides by 100, we can eliminate it:
0.46 = (1/2)^(t / T)
To isolate t, we can take the logarithm of both sides (base 1/2):
log(0.46) = (t / T) * log(1/2)
Now, we can solve for t by multiplying both sides by T and dividing by log(1/2):
t = T * (log(0.46) / log(1/2))
Calculating these values will give you the answers to the questions.