I asked the following the other day and received the following answer:
Posted by Matthew on Sunday, March 17, 2013 at 11:03pm.
A quantity of oxygen gas had 16.32 g of the radioactive isotope oxygen-19 in it. When measured exactly 10 minutes later, the amount of oxygen-19 was 0.964 g. What is the half-life, in seconds, of oxygen-19?
math - Reiny, Monday, March 18, 2013 at 12:05am
16.32(1/2)^(t/k) = .964 , where t is in minutes, and k is the half-life in minutes.
(.5)^(10/k) = .0590686.. ( I stored it)
ln both sides
(10/k) ln .5= ln (.059068...)
10/k = 4.081464...
k = 2.4504 minutes or 147.006 seconds
I would like to know
1) How does this mathematically work, how can i ln both sides suddenly?
2) Secondly, how does the last answer occur- like this part is not giving me the same outcome:
10/k = 4.081464...
k = 2.4504 minutes or 147.006 seconds
nevermind on 2), i would appreciate an asnwer to 1) though
log( ? ) is a mathematical operation, just like taking a square root, cubing something of finding the sine of something
Basic rules of an equation: Whatever you do to one side, you must do to the other side.
so if I have
.5^(10/k) = .059..
my doing
LOG .5^(10/k) = LOG .059...
I am doing just that, whatever I did to the left side, I did to the right side.
If you don't know logarithms (logs), then there is no way you can solve this equation, other than plain old trial and error guessing
but why the choice for ln?
1) ln stands for the natural logarithm function. In this specific case, we need to take the natural logarithm of both sides of the equation to isolate the variable in the exponential term. By applying the natural logarithm, we can undo the exponential function on one side of the equation.
When we take the natural logarithm of both sides of the equation 16.32(1/2)^(t/k) = 0.964, we get ln(16.32) + ln((1/2)^(t/k)) = ln(0.964).
Using the properties of logarithms, we can simplify the equation further. The property we use here is ln(a^b) = b * ln(a).
So the equation becomes ln(16.32) + (t/k) * ln(1/2) = ln(0.964).
The reason why we can suddenly take the natural logarithm on both sides is because the natural logarithm function is the inverse of the exponential function. By using the natural logarithm, we are essentially undoing the effect of the exponential function, which allows us to solve for the variable inside the exponent (in this case, t/k).
2) To solve for k, we rearrange the equation ln(.5)(10/k) = ln(.0590686...) to isolate k on one side.
We divide both sides of the equation by ln(.5), and we get (10/k) = ln(.0590686...)/ln(.5) (approximately 4.081464...).
To solve for k, we divide both sides by 10 to isolate k on one side, resulting in 10/k = 4.081464..., as you mentioned.
Finally, to find the value of k in seconds, we convert the value of k from minutes to seconds. Since there are 60 seconds in a minute, we multiply 2.4504 minutes by 60, which gives us 147.024 seconds (rounded to 147.006 seconds in the answer you received).