One method of estimating the thickness of the ozone layer is

One method of estimating the thickness of the ozone layer is to use the formula

ln I₀ ln I = kx, where I₀ is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength.

Approximate the percentage decrease in the intensity of light with a wavelength of 3176 × 10⁸ centimeter with k 0.39 if the ozone layer is 0.21 centimeter thick. (Round your answer to two decimal places.)

I know I0/I= 1.085 but am unsure where to go from there. Please show the work. Thank You!

Solution

ln(Io) -ln(I) = kx

where I0 is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength.

k = 0.39 ; Io = 3176 x 10^-8 cm = 0.00003176 cm ;x = 0.21 cm Find I

So,ln(3176x10^-8)- lnI = 0.39*0.21

ln3176 + ln (10^-8) - lnI = 0.0819

8.063 - 18 .42 - lnI = 0.0819

lnI = -10.4389

I = 0.00002927138 cm

Now percentage change in wavelength =( 0.00003176 - 0.00002927138)/ 0.00003176

= 0.0783*100 = 7.8 % decrease in wavelength

Io/I = 0.00003176/0.00002927138

= 1.085

I = Io/1.085 = 0.921Io

% decrease = {(Io - 0.921Io)/Io}*100 = 7.83% decrease