# Math Help - [SOLVED] isotopic decay - natural log and straight lines

1. ## [SOLVED] isotopic decay - natural log and straight lines

The question starts off telling us that a laboratory experiments a quantity of a radioactive isotope, which was placed in a closed container.

Measurements were taken to see how much of the isotope remained after various intervals.

The experimenter did not provide information on the initial mass 0 M of the isotope, but suggested that M and 0 M were related by the decay equation
M = M e−kt where k is the decay constant.

(a) asked to transform the equation by taking natural logarithms so that a striaght line graph of the form y=mx+c may be plotted. Also state how y,m and c are related to M, M_0, k and t.
I managed to get
ln(M)=ln(M_0*e^(-kt))

ln(M)=ln(M_0) + lne(^(-kt))

ln(M)=ln(M_0) -kt

so y=lnM
m= -k
x= t
c=lnM0

(b)however in the 2nd question, it asks to use my transformation in part (a) to make a new table of values and plot a graph...
This is the part i'm stuck on From the equation y=mx+c, I should have InM=-kt+InM_0

However I'm unsure of how to use the values in the table given in the image, to make a new table. If t(hrs) = 5, initially M(mg) was 21.489. But now I have to make a new table with y=mx+c, and i dunno how to input the values into the equation i just made help in the right direction please?

2. I'm going to change up the notation you're using just to make it a bit more clear. Instead of M_0 for the mass at time zero, I'm going to use I for the initial mass.

The given decay equation is then $M=I*e^{-kt}$

You've done a fine job with the logrithm to rearrange this into $ln{M}=-kt+ln{I}$

The table lists the independent variable and the dependent variable. Your new table should do this as well. Thus, since your 'independent variable' is now $t$ you can list it in the new table (just as was done in the original data). Since your 'dependent variable' is $ln{M}$ you should take the natural log of each mass in the original table and list that result in the new table.

This sort of problem comes up quite often. Another way to get a linear graph of this data is to plot t versus M on graph paper specifically designed for this kind of data (called semi-log paper), in which the y-axis has a logarithmic scale.