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Exponential Growth and Decay

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Exponential Growth and Decay

Postby Chemhalp » Wed Apr 23, 2008 6:11 pm

So I have been given a couple of questions from my tutor however I have no idea where to start, I'm not asking for the answers just a push in the right direction.

1)

270 Vibrio cholerae cells were inoculated into 10 ml of a liquid culture and incubated for 9 hours. At the end of this time the cells (in a volume of 100 microlitres) were spread onto agar plates and the colony forming units were enumerated after overnight incubation by counting the colonies on plates from different dilutions: e-5 = 643; e-6 = 75; e-7 = 6

Assuming that the cells multiplied by binary fission and the culture was in synchronized exponential growth throughout the incubation, how many full generations did the culture undergo and what was the generation time.

I guessed that you can only use one of the counts from the dilutions to perform the calculations and the number should be between 30 and 300 so I guess i'm using the 10^-6

2)

500 microlitres of a solution of a DNA fragment labelled with 32P with an initial activity of 0.4 kBq in 10 ml was needed for an experiment 9 days later. Calculate the radioactivity of the sample used in the experiment (T = half-life)

(I think I first need to work out the amount of radioactivity in the sample to be used in the experiment, but couldn't work that one out)

3)

Barium-140 has a half light of 13 days. How many decay half-lives does a sample of barium-140 undergo in 6 weeks?
What fraction of the sample would remain after 6 weeks?

(Confusing question, I guessed barium-140 would go through 3.23 half-lives and that it would be around 1/8th of the sample left over, but exact number eludes me)
Last edited by Chemhalp on Wed Apr 23, 2008 7:59 pm, edited 1 time in total.
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Postby mith » Wed Apr 23, 2008 6:49 pm

Amount of original stuff * (1/2)^n = stuff left

where n is number of half lives.

You can derive a similar equation for calculating number of half lives.
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Re: Exponential Growth and Decay

Postby Chemhalp » Wed Apr 23, 2008 7:58 pm

So for question three,

42 days / 13 day half life = 3.23
ergo
0.5^3.23 = 0.106579361, which is fraction of the sample after the 6 weeks?
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Postby mith » Wed Apr 23, 2008 10:27 pm

Well does the answer make intuitive sense?

what is happening when you raise 1/2 to a power? You can write out the integer series if it helps.
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Postby Chemhalp » Thu Apr 24, 2008 4:03 pm

Yeah it makes sense, it should be around an 8th (0.125) anyway so I guess its right.

Can I use the formula you gave for all of the questions?
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Postby mith » Thu Apr 24, 2008 7:51 pm

http://en.wikipedia.org/wiki/Exponential_growth

If you manipulate the formula a little, its basically what you'll need for exponential stuff.
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Re: Exponential Growth and Decay

Postby Chemhalp » Thu Apr 24, 2008 9:49 pm

Ok just had a look at question 1 again...

270 cells in 10ml then 100 microlitres taken

So... 10000/100 = 100 so 270/100 = 2.7

So... starting with 27 and ending with 75 is what I have right?

original stuff * (1/2)^n = stuff left

as you said above

So... 2.7 * 0.5^(n) = 75

75/2.7 = 0.5^(n)

But that gives me like 0.5^n = 27 which can't be...

So then I got thinking that its at dilution e-6 so something else should have been done -_-

Any hints?
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Postby mith » Thu Apr 24, 2008 10:32 pm

note that when you multiple by 1/2 ^ n , it halves it n times.

In this case you're looking at growth, specifically binary fission so you should be doubling n times i.e. 2^n.
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Postby Chemhalp » Thu Apr 24, 2008 11:25 pm

ah yeah, exponential growth not decay

Cheers,

So its 75/2.7 = 2^n, right

2^n = 27.7777778

How do I go on from there?
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Postby Chemhalp » Sat Apr 26, 2008 12:10 am

log(x^n) = nlog(x).

right right?

or log(27.7777778) / log(2) = 4.79585928

Right?

I can't tell if i've done question 1 right at all :(
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Postby mith » Sat Apr 26, 2008 1:21 am

I'm not quite sure but from the way you wrote it, it seems like this is how it looks

You have a dilution with "x" number of colonies and it multiplied "n" times

x* 2^n=6

for the second dilution

10x * 2^n = 75

from third

100x * 2^n =643

But it's impossible to solve for the x or n because all 3 equations are linearly dependent...so I'm not sure what additional information I'm missing...
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Postby Chemhalp » Sat Apr 26, 2008 10:11 pm

or...

75 is the number of colonies and not the number of cells so you have to divide the number of colonies by the dilution factor which was 10^-6 and then x 100 to get the number of cells in 10ml. I got 7.5x10^9.

Then ln(7.5x10^9/270) /ln2 = 24.7
so the number of divisions is 24

maybe?
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