Word Problem with Constant 'e'

[darshanpatel]

Homework Statement

For living organic material, the ratio of the number of radioactive carbon isotopes (carbon-14) to the number of non-radioactive carbon isotopes (carbon-12) is about 1 to 10^12. When organic material dies, its carbon 12 content remains fixed whereas its radioactive carbon-14 bagins to decay with a half-life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula which denotes the ratio of carbon-14 to carbon-12 present at any time 't' (in years).

R=(1/10^12)e^(-t/8223)

Estimate the age of newly discovered fossil in which the ratio of carbon-14 to carbon-12 is 1/10^13

Homework Equations

-None-

The Attempt at a Solution

I just substituted one thing, I got confused, do I plug in 5,700 for 'R' or 11,400 for 'R' then solve?

From this: R=(1/10^12)e^(-t/8223)
To this: R=(1/10^13)e^(-t/8223)

That is as far as I got

Please help! Thanks

 

[Simon Bridge]

You have to understand what the different terms mean.
The general equation would be:
R=R_0e^{-t/\tau} where
R_0 is the ratio that it started out with (at t=0);
\tau is the "mean-life" of the process in question, related to the half-life[1] by \tau=T_1/2/ln(2); and
R is the ratio now (i.e. after some time t has passed).

eg. for C14, the half-life is 5700yrs so the mean life is 8223yrs. So \tau=8223\text{yrs}.

So - is 1/10^13 the ratio it started with or the ratio now?
Where does it go?

----------------------------
note: Also written R=R₀e^λt
... which should look familiar if you ever played "hλlf life" :)

λ = ln(2)/T_1/2

 

[darshanpatel]

Yeah I don't understand any of what you just said. I know you replace the 1/10^12 to 1/10^13 because they gave you a new ratio. But for what everything is equal to, 'R,' do I put in 5700 or what?

 

[Simon Bridge]

Yeah I don't understand any of what you just said.

If you don't at least try you are going to fail your course!

I know you replace the 1/10^12 to 1/10^13 because they gave you a new ratio.

This is not correct.

But for what everything is equal to, 'R,' do I put in 5700 or what?

The answer is "what". Please be aware that I am not going to do your homework for you. You have to do it yourself.
So you have to figure out for yourself what to put for R, but I can guide you.
You figure it out by understanding what it means - what does R, in the equation stand for? What does it mean?
What kind of thing is the 5700years? What is it called?

Consider what the equation is telling you:
If the creature had died today, then the ratio would be 1/10^12
In 5700 years the ratio would have been 0.5/10^12 because e^(5700/8223)=0.5

This also means that if the creature had dies 5700 years ago, the ratio back then would have been 1/10^12 and now it will read as 0.5/10^12.

Get it yet?

 

[darshanpatel]

Oh, I think I get it.. So for the thing to decay fully we plug 11400 into 't' because it said it began to decay with a half-life of about 5700 years.

Once we plug it in, we can put it in the calculator and found out what 'R' is right?

I am saying I do this:

R=(1/10^13)e^(-11400/8223)
?

 

[Simon Bridge]

Please answer the questions.
What does R stand for in the equation?
What does t stand for in the equation?

What is the quantity you are trying to find?

Also please answer the questions in post #4.
Those are not rhetorical - they are put there for you to tell me the answers.

 

[darshanpatel]

What I said in thread starter post, is all the problem gives me, that's why I am confused just like you.. Sorry.

All I can answer for you is that 't' stands for time in years only because that is given.

 

[Simon Bridge]

Time is correct for t.

The answer to everything I asked you is in post #1 and post #2.

Do you even know what the point of the equation is?
My problem is that in order to guide you, I need to find some point in your course where you do understand something. But, if I take what you have said at face value, you don't even understand basic algebra. I'll see if I can get someone more experienced to see if they can figure out what to do.

 

[darshanpatel]

Can you please show me what the formula would look like with everything substituted in? Maybe then I might understand

 

[Simon Bridge]

Nice try: that sort of thing has a special name, it is called "doing your homework for you". We don't do that :)

I did give you some examples in post #4

Consider what the equation is telling you:
If the creature had died today, then the ratio would be 1/10^12
In 5700 years the ratio would have been 0.5/10^12 because e^(5700/8223)=0.5

This also means that if the creature had died 5700 years ago, the ratio back then would have been 1/10^12 and now it will read as 0.5/10^12.

OK - so let's do the last one:
A body is found that has a C14 ratio of 0.5/10^12
How long ago did it die?

We reason that if the current C14 ratio is R, then t will be the time since the C14 ratio was at maximum. (post #2) We write\frac{0.5}{10^{12}} = \frac{1}{10^{12}}e^{t/8332}then solve for t.

 

[Redbelly98]

To estimate the age of dead organic material, scientists use the following formula which denotes the ratio of carbon-14 to carbon-12 present at any time 't' (in years).

R=(1/10^12)e^(-t/8223)

Okay, they are saying that R:

(1) is equal to the expression above,

and

(2) is the ratio of carbon-14 to carbon-12 present at any time 't' ...

Estimate the age of newly discovered fossil in which the ratio of carbon-14 to carbon-12 is 1/10^13

And here they are telling us that at some time 't', the ratio of carbon-14 to carbon-12 is equal to 1/10¹³.

In other words:

R is the ratio of carbon-14 to carbon-12,​

and

the ratio of carbon-14 to carbon-12 is 1/10¹³.​

Therefore, R = _____?

 

[darshanpatel]

would i put 5700 in for 't' and change 1/10^12 to 1/10^13?

Then solve for R?

 

[Redbelly98]

would i put 5700 in for 't' and change 1/10^12 to 1/10^13?

Then solve for R?

No.

There is a simple, logical connection you have to make here:

R is the ratio of carbon-14 to carbon-12,​

and

the ratio of carbon-14 to carbon-12 is 1/10¹³.​

Therefore, R = _____?

Fill in the blank, use this value for R, then solve for t.