You have 20 grams of phosphorus-32 that decays 5% per day. How long will it take for hlaf of the original amount to decay?
This may not be the math process you want. From a chemists standpoint, I would do the following.
ln(No/N) = kt. We must determine k.
So at zero time we start with 20 g. At the end of 1 day we have 20g-5% = 19 g. Plug that into the first order equation of ln(20/19) = k(1 day)
k = 0.0513
Then ln(20/10) = 0.0513(t)
Solve for t. If I didn't goof that is
13.51 days which I would round to 13.5 days.
Well, well, well, looks like we've got ourselves a decaying situation here! So, you've got 20 grams of phosphorus-32, and it decays by 5% every day. To figure out when half of the original amount decays, we can use the concept of half-life.
Now, with a decay rate of 5% per day, it means that after one day, you'll have 95% of the original amount left (since 100% - 5% = 95%). So, the remaining amount after one day would be 0.95 * 20 grams.
To find out how long it takes for half of the original amount to decay, we can set up an equation where the remaining amount is equal to half of the original amount (which is 10 grams):
0.95^t * 20 grams = 10 grams
Solving this equation will give us the time it takes for half of the original amount to decay. However, since I'm a Clown Bot, I tend to avoid doing math and instead focus on making you laugh. So, go ahead and grab a calculator, solve that equation, and I'm sure you'll have a blast!
To determine how long it will take for half of the original amount of phosphorus-32 to decay, we need to use the concept of a half-life.
A half-life is the amount of time it takes for half of a radioactive substance to decay.
Given that phosphorus-32 decays at a rate of 5% per day, we can calculate its half-life using the formula:
Half-life = (ln(2)) / (decay constant)
First, let's calculate the decay constant. The decay constant is the natural logarithm of 2 divided by the percentage decay per day:
Decay constant = ln(2) / (5/100)
Now we can calculate the half-life:
Half-life = ln(2) / (ln(2) / (5/100))
Simplifying the equation:
Half-life = ln(2) / (ln(2) * (100/5))
Canceling out the natural logarithms:
Half-life = 1 / (100/5)
Half-life = 1 / 20
Therefore, the half-life of phosphorus-32 is 1/20th of a day.
To determine how long it will take for half of the original amount to decay, we can multiply the half-life by the number of half-lives:
Time to decay = Half-life * Number of half-lives
In this case, since we want to know when half of the original amount decays, the number of half-lives would be 1.
Time to decay = (1/20) * 1
Time to decay = 1/20 day
So, it will take 1/20th of a day for half of the original amount of phosphorus-32 to decay.
I looked up the half life of P-32. It is 14.28 days. I suspect 5% decay in the problem is just a close number.
Using the "continuous" decay formula, I got
.5 = 1(e^(-.05t) )
ln .5 - -.05t lne, but ln e = 1
t = -.05/ln .5 = .072134.. years
= 26.3 days