Radioactive Decay

Part 1. Background Information

Suppose  y  is a function of the form  y=y0*b^t  with    b<1.   Explain here why the time required for the value of  y  to become half its starting value is independent of the starting value.

Part 2. Rapid Decay

The Meyer-von Schweidler data:

\(activity\  = 
      \ {\ {0.2, \ 35.0}, \ {2.2, \ 25.0}, \ {4.0, \ 22.1}, \ 

        		{5.0, \ 17.9}, \ {6.0, \ 16.8}, \ {8.0, \ 13.7}, \ 

        		{11.0, \ 12.4}, \ {12.0, \ 10.3}, \ {15.0, \ 7.5}, \ 

        			{18.4, \ 4.9}, \ {26.0, \ 4.0}, \ {33.0, \ 2.4}, \ 

        			{39.0, \ 1.4}, \ {45, \ 1.1}\ }; 

    ListPlot[activity, \ Prolog -> PointSize[.02]]\)

Respond to Step 1 questions about the data here.

The next command loads a "package" of plotting routines, one of which is a semilog routine called LogListPlot.  It works in much the same way as ListPlot, but it does logarithmic scaling of the vertical axis.

<< Graphics`Graphics`;

logact  =  LogListPlot[activity,  Prolog -> PointSize[.02]]

Type answers to questions in Step 2 here.

Enter command lines below to calculate coefficients for a line  y=m*t+c  that should fit the semilog plot reasonably well.  Note that Mathematica's command is Log[b,z] for the log of z base b.

Explain your fitting procedure here.

Activate the following command to see your line, and compare this plot with the semilog plot in Step 2.  If the line does not look approximately like the data plot, rework Step 3.

Plot[m*t + c,   {t,  0,  45}]

Calculate the constants for a model function of the form  y=y0*b^t.  Use the names  y0  and  b  for these constants.

The following commands will display your fitted function with the data.  (The DisplayFunction operator is set to "nothing" by using the word Identity to keep the graph from being displayed.  The DisplayFunction is then reset the the "usual operator" called $DisplayFunction to see the result on screen.)

\(fitplot\  = \ 
      LogPlot[y0*b^t, \ {t, \ 0, 45}\ , 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> Identity]
        \)\(Show[fitplot, \ logact, \ 
	\ \ \ \ \ \ \ \ \ \ \ 
      DisplayFunction -> $DisplayFunction]\)

Now display the fitted function with the data in a Cartesian plot.

\(fitplot2\  = \ 
      Plot[y0*b^t, \ {t, 0, 45}, \ 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        DisplayFunction -> Identity]\)\(activity2\  = \ 
      ListPlot[activity, \ Prolog -> PointSize[.02], \ 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        DisplayFunction -> Identity]\)\(Show[fitplot2, \ activity2, \ \ 
	\ \ \ \ \ \ \ \ \ \ \ 
      DisplayFunction -> $DisplayFunction]\)

Part 3. The Half-Life

Do your half-life calculations here.

Write your discussion and explanation here.

Part 4. Summary