1. ## Using natural logoritums

Ok, here is my question

In a laboratory experiment a quantity of a radioactive isotope was placed in a closed container. Measurements were taken to determine how much of the isotope remained after various intervals of time. The experimenter omitted to provide information on the initial mass M0 (0 is small at bottom of M) of the isotope, but postulated that M and M0 were related by the decay equation:

M = M0e-kt (-kt is small and at the top, as if its to the power of -kt).

where k is the decay constant and t is time. Show how to transform the equation by taking natural logarithms so that a straight line graph of the form y = mx + c may be plotted, and state how y , m and c are related to M , M0 , k and t .

2. Originally Posted by oli212
Ok, here is my question

In a laboratory experiment a quantity of a radioactive isotope was placed in a closed container. Measurements were taken to determine how much of the isotope remained after various intervals of time. The experimenter omitted to provide information on the initial mass M0 (0 is small at bottom of M) of the isotope, but postulated that M and M0 were related by the decay equation:

M = M0e-kt (-kt is small and at the top, as if its to the power of -kt).

where k is the decay constant and t is time. Show how to transform the equation by taking natural logarithms so that a straight line graph of the form y = mx + c may be plotted, and state how y , m and c are related to M , M0 , k and t .

You have a decay equation:

$\displaystyle M(t)=M_0 e^{-kt}$

Now take logs:

$\displaystyle \ln(M)=\ln(M_0)-kt$

So put $\displaystyle y=\ln(M)$, ...

RonL

3. Ok thanks,

So just to confirm,

Y would = ln (M)
M would = ln(M0)
and C would = t

????

4. Originally Posted by oli212
Ok thanks,

So just to confirm,

Y would = ln (M)
M would = ln(M0)
and C would = t

????
No $\displaystyle y$ is $\displaystyle \ln(M)$, $\displaystyle x$ is $\displaystyle t$, $\displaystyle m$ is $\displaystyle -k$, and $\displaystyle c$ is $\displaystyle \ln(M_0)$

RonL

5. Ah i see.

Is there any chance you could just quickly run through how you got there?

6. Also, the question asks to estimate the value of the decay constant k.

What would that be?

7. Originally Posted by oli212
Also, the question asks to estimate the value of the decay constant k.

What would that be?

You plot a curve of $\displaystyle \ln(M)$ against $\displaystyle t$. The slope
of the line of best fit is the estimate of $\displaystyle -k$

RonL