SOLUTION: beryllium-11 decomposes into boron-11 with a half life of 13.8 seconds. How long will it take 240 g of beryllium-11 to decompose into 7.5g of beryllium-11? based on the forumu

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: beryllium-11 decomposes into boron-11 with a half life of 13.8 seconds. How long will it take 240 g of beryllium-11 to decompose into 7.5g of beryllium-11? based on the forumu      Log On


   

Question 253874: beryllium-11 decomposes into boron-11 with a half life of 13.8 seconds. How long will it take 240 g of beryllium-11 to decompose into 7.5g of beryllium-11?
based on the forumula provided in the text book C(t)=C(0)2e^-t/h
where C(0) = 240, C(t)=7.5 ,h=13.8
7.5/240=2e^-t/13.8
this was a s far as i got
Found 3 solutions by ankor@dixie-net.com, Theo, martin.jabou:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
beryllium-11 decomposes into boron-11 with a half life of 13.8 seconds.
How long will it take 240 g of beryllium-11 to decompose into 7.5g of beryllium-11?
:
Here is what it should be then:
2^(-t/13.8) = 7.5%2F240
2^(-t/13.8) = .03125
:
Find the natural log of both sides
ln(2^(-t/13.8)) = ln(.03125)
:
log equiv of exponents
-t%2F13.8*ln(2) = ln(.03125)
:
-t%2F13.8(.693) = -3.4657
:
-.693t%2F13.8 = -3.4657
:
Multiply both sides by -13.8
.693t = -13.8 * -3.4657
693t = +47.827 seconds
t = %2847.827%29%2F%28.693%29
t = 69 sec
:
:
Check on a calc: enter 240*2^(-69/13.8) = 7.500

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your equation to work with is:

C%5Bt%5D=C%5B0%5D%2A2%2Ae%5E%28-t%2Fh%29

The exponent in the equation is -t%2Fh.

C%5B0%5D+=+240
C%5Bt%5D+=+7.5
h+=+13.8

substituting in that equation, you get:

7.5+=+240+%2A+2+%2A+e%5E%28-t%2F13.8%29

the exponent in the equation is -t%2F13.8%29

you want to isolate the exponential term on the right hand side of the equation so you need to divide both sides of the equation by 240 * 2 to get:

7.5%2F%282%2A240%29+=+e%5E%28-t%2F13.8%29

the exponent in the equation is -t%2F13.8%29

take the log of both sides of this equation to get:

log%287.5%2F%282%2A240%29%29+=+log%28e%5E%28-t%2F13.8%29%29

since log(b^c) = c*log(b), your equation becomes:

log%287.5%2F%282%2A240%29%29+=+%28-t%2F13.8%29+%2A+log%28e%29

multiply both sides of this equation by 13.8 to get:

13.8%2Alog%287.5%2F%282%2A240%29%29+=+-t%2Alog%28e%29

divide both sides of this equation by -log(e) to get:

t+=+%2813.8%2Alog%287.5%2F%282%2A240%29%29%29%2F%28-log%28e%29%29

solve for t to get:

t+=+-24.92528364%2F-.434294482+=+57.39258655 seconds.

substitute in original equation to see if this value for t is good.

original equation is:

C%5Bt%5D=C%5B0%5D%2A2%2Ae%5E%28-t%2Fh%29

exponent in the equation is -t%2Fh

C%5B0%5D+=+240
C%5Bt%5D+=+7.5
h+=+13.8
t = 57.39258655

equation becomes:


7.5=240%2A2%2Ae%5E%28-57.39258655%2F13.8%29

exponent in the equation is %28-57.39258655%2F13.8%29

simplify to get:

7.5=480%2Ae%5E%28-4.158883083%29

simplify further to get:

7.5+=+7.5

since the equation is true, the value for t is good.

t = 57.39258655 seconds

Answer by martin.jabou(1) About Me  (Show Source):
You can put this solution on YOUR website!
The solution is quite simple actually:
You know you are going from 240g of Be-11 to 7.5g of Be-11, so write down what you have -
Half Life Formula is C(t) = Cø2^(-t/H)
C(t) = 7.5g, Cø = 240g, H = 13.8 seconds
Sub In:
7.5 = 240 (2)^(-t/H)
Divide both sides by 240 to get:
7.5 / 240 = 2^(-t/13.8)
Simplify:
1 / 32 = 2^(-t/13.8)
1 / 32 must be written as a base two: therefore it becomes 2 ^ -5, your new equation is:
2^-5 = 2^(-t/13.8)
Since at this point you have the same base 2 on both sides, you can now equate the exponents to each other and solve for variable t. You get:
-5 / 1 = -t/13.8
-69 = - t, divide each side by -1 (not particularly necessary since we know time cannot be negative) and we get T is equal to 69 seconds :)