Analyzing Data

Introduction

In certain cases you might expect your experimental data to follow either an exponential model, y = A Exp[r x], or a power model y = A x^k. One can test the data for these models using logarithms.

For the exponential model taking the logarithm of both sides gives Log[y] = Log[A] + r x. A plot of the Log[y] versus x gives a straight line with intercept Log[A] and slope r. (Or one could plot the raw data on semi-log paper.)

For the power model taking the logarithm of both sides of the equation gives Log[y] = Log[A] + k Log[x]. A plot of Log[y] versus Log[x] gives a straight line with slope k and intercept Log[A] if the data follows the power model. (Or one could plot the raw data on log-log paper.)

Barium-137 Data

Barium-137 is an unstable isotope.The following data was collected by Professor Clifton Albergotti, USF Physics Department. The first number in each data pair is time in minutes, and the second number is a measure of radioactivity for the sample. The following example illustrates one possible approach to finding the half life of this isotope.

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The data looks like exponential decay. If we try to do a linear fit of the data we get a terrible result.

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If we take the logarithm of the "y" values and replot the data we get a pretty straight line.

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Doing a linear fit of the transformed data works well. This is strong evidence that the original data was exponential decay.

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We can exponentiate our results, and we have a nice exponential fit of our data. These manipulations were necessary because most calculators and computer programs are unable to do an exponential fit of the data directly (or if they can, then they do what we just did).

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Half-life

Now we can easily calculate the half life. First find the radioactivity at t=0:

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Now solve for t when the radioactivity is half of this amount:

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Or you could use the formula from a textbook that gives the half life in terms of the rate constant as -Log[2]/(-k)

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The half life is about 2.77 minutes. (The original data contained at most four significant figures, therefore calculated results based on that data should have no more than four significant figures. Since computers use much greater precision, it is necessary to adjust final results appropriately.)