Calculations Using Beer's Law

Recall that Beer's Law can be written as: I z = I 0 * exp -k*z

where I z is the light intensity at depth z
where I 0 is the light intensity at the surface
where k is the attenuation coefficient

In this exercise, we said I 0 = 1500 microEinsteins per meter squared per second and we applied two different values for k, 0.02 and 0.04 to compute I z at different depths in the water column.

The best way to set up these types of problems is to use a spreadsheet program like Excel. Here's what your spreadsheet would look like to calculate values for this particular problem. It looks more complicated than it is because all you need to do is type the correct equation in cell B2 and C2 and then use FILL DOWN for all the rest. Note that the value of k is fixed as cell $B$1 or $C$1. The value for I 0 is entered as the actual number but you could create a fixed cell for this variable and examine different values of I 0. Values for depth in column A are allowed to vary independently in both columns B and C, as you can see.

A B C
1 Depth 0.02 0.04
2 0 =(1500)*EXP(-$B$1*A2) =(1500)*EXP(-$C$1*A2)
3 20 =(1500)*EXP(-$B$1*A3) =(1500)*EXP(-$C$1*A3)
4 40 =(1500)*EXP(-$B$1*A4) =(1500)*EXP(-$C$1*A4)
5 60 =(1500)*EXP(-$B$1*A5) =(1500)*EXP(-$C$1*A5)
6 80 =(1500)*EXP(-$B$1*A6) =(1500)*EXP(-$C$1*A6)
7 100 =(1500)*EXP(-$B$1*A7) =(1500)*EXP(-$C$1*A7)
8 120 =(1500)*EXP(-$B$1*A8) =(1500)*EXP(-$C$1*A8)
9 140 =(1500)*EXP(-$B$1*A9) =(1500)*EXP(-$C$1*A9)
10 160 =(1500)*EXP(-$B$1*A10) =(1500)*EXP(-$C$1*A10)
11 180 =(1500)*EXP(-$B$1*A11) =(1500)*EXP(-$C$1*A11)
12 200 =(1500)*EXP(-$B$1*A12) =(1500)*EXP(-$C$1*A12)

If you entered everything correctly (or if you did it by hand), you should have computed a table of numbers very close to the values below.

Depth
(meters)
k=0.02 k=0.04
0 1500.00 1500.00
20 1005.48 673.99
40 673.99 302.84
60 451.79 136.08
80 302.84 61.14
100 203.00 27.47
120 136.08 12.34
140 91.22 5.55
160 61.14 2.49
180 40.99 1.12
200 27.47 0.50

As you can see, small changes in k cause big changes in the intensity of light at a particular depth. In the first calculation (k = 0.02), the depth of the euphotic zone (where I z = 0.01 *  I 0 or 15 microEinsteins per meter squared per second) is below 200 meters. In the second calculation, the euphotic zone ends somewhere between 100 and 120 meters.