3. The fraction F of light that is absorbed by any gas in air is logarithmically

Question

3. The fraction F of light that is absorbed by any gas in air is logarithmically related to the concentration c of the gas and the distance d through which the light travels; this relationship is called the Beer-Lambert law: IJ In ( 1-F) =-Ked Here K is a proportionality constant. Show by simple trial calculations that for concentrations near zero (e.g., where Ked = 0.001), that F is related almost linearly to c, whereas for larger Ked valucs (c.g., ncar 2), that doubling the concentration does not nearly double the light absorption.

Explanation / Answer

Let Kcd=x

and then we have the equation as

ln(1-f)=-x

for x=0.001

f=1-e^(-0.001)=0.00099==0.001(== represents approximately)

for x=0.002

f=1-e^(-0.002)=0.00199==0.002

for x=0.003

f=1-e^(-0.003)=0.00299==0.003

for x=0.004

f=1-e^(-0.004)=0.00399==0.004

Hence from the above values we can infer that when Kcd is small f varies linearly

But

for x=1

f=1-e^(-1)=0.632

for x=2

f=1-e^(-2)=0.864

Here from this we can observe as f at x=2 is not the double of x=1 hence it doesn't follow a linear relationship for large values of Kcd (i.e. x).

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