Half Life of 14C

Purpose

To determine the half life of 14C from experimental data of the number of remaining 14C atoms as a function of time.

Objectives

• To learn how to use spreadsheets for recording and analyzing data
• To learn how to plot on a log scale in order to straighten exponential functions
• To learn how to use linear regression for the best fit of data
• To learn how to represent physical data using mathematical and statistical methods.

Theory

The number of radioactive atoms decreases with time  following an exponential behavior. The time for which the number of remaining radioactive atoms is decreased to ½ of its original value is called Half-Life, .

For 14C, the established half-life period is 5730 ± 40 yrs.

For radioactive atoms, the number of remaining radioactive atoms as a function of time is given by the function:

                Eq. 1

Here  is the decay constant and it is related to the half-life period,  as:

                Eq. 2

When plotted on a linear scale, the graph for the function of the number of particles N(t) is exponential. The best way to verify that that graph is exponential is to plot the function on a logarithmic scale: ln(N) versus t. In that case, the graph should appear as a straight line with a slope of the best line fit expected to be equal to , since

                        Eq. 3

Simulated Experimental Data

The file below contains simulated data of the radioactive decay of 14C. The first column gives the time in years elapsed since the beginning, and the second column gives the number of remaining 14C atoms.

https://docs.google.com/spreadsheets/d/1TQWnPr3QqRxCOtrDNScqkwVuqd_H-Cb4AP9u0tT9H10/edit?usp=sharing

Activity 1. Plot N vs. T

Import that experimental data into a spreadsheet and plot N vs. T (yr).

Question: How can you tell that the graph is truly exponential and not some other curve?

Activity 2. Plot Ln N vs. T

Create a column of data in the spreadsheet with the natural log values of N. Plot ln(N) versus T in yr.

Question: Does the graph appear like a straight line?

Activity 3. Determine the half-life from the the Ln N vs. T graph.

Using Linear regression, determine the best line through your radioactive data and its slope.

The slope of that graph should be   (see Eq. 3)

Determine the half-life period of 14C in years (see Eq. 2)

Compare your result for the half-life period in years with the established value of 5370 ± 40 yrs.