Additional hints for worksheet 4.1: How to construct an absorption photometer on your own (4/4)

We start with the transmission of light over the distance of x,

T= I(x) I o = e −ax

and differentiate it with respect to the absorption coefficient:

dT da =−x  e −ax =−xT

Now we interpret the differentials da and dT as an error Δa of the calculated absorption coefficient which is the result of the measurement error ΔT of the transmission. From this, for the absolute absorption coefficient, it follows:

Δa=− 1 x ΔT T =− e ax x ΔT

The relative error becomes:

Δa a =− 1 ax ΔT T =− e ax ax ΔT

In order to find a possible minimum of the relative error, we differentiate it with respect to the absorption coefficient a,

d da ( e ax ax )ΔT=( x e ax ax − e ax a 2 x )ΔT

and equate the derivate with zero. After reducing, the following is obtained:

ax=1

The measuring distance which shows the smallest error in the calculated absorption coefficient hence is numerically equal to the inverse absorption coefficient:

x= 1 a

If inserted in the equation for the transmission, it follows:

T= I(x) I o = e −a/a = 1 e =0.368...

The best conditions for precise measurements are given when the intensity of light has declined through absorption to around 37% of the starting value.

Task: Relative and absolute errors of the absorption coefficient ↓