Proof: why absorbance value of spectrophotometer cannot be greater than 2
Spectrophotometer:
The spectrophotometer is an instrument which measures an amount of light that a sample absorbs.
The spectrophotometer works by passing a light beam through a sample to measure the light
intensity of a sample. The working principle of spectrophotometer is represented by Figure 1.
Figure 1: Principle of Spectrophotometer
Where, 𝐼0 is Incident light and 𝐼 is transmitted light.
Transmittance is the ratio of the transmitted intensity I, over the incident intensity I0,
𝑻𝒓𝒂𝒏𝒔𝒎𝒊𝒕𝒕𝒂𝒏𝒄𝒆 = 𝑇 =
𝐼
𝐼0
%𝑇𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑎𝑛𝑐𝑒 = %𝑇 = 100𝑇
(1)
(2)
Whereas the absorbance has a logarithmic relationship to the transmittance,
𝑨𝒃𝒔𝒐𝒓𝒃𝒂𝒏𝒄𝒆 = 𝐴 = 𝑙𝑜𝑔10
𝐼0
𝐼
(3)
From Eq. (1)
𝑇=
𝐼
𝐼0
𝐼0
1
=
𝐼
𝑇
⇨
(4)
So, Absorbance may be written as,
𝐴 = 𝑙𝑜𝑔10
1
𝑇
(5)
𝐴 = 𝑙𝑜𝑔10
100
%𝑇
(6)
𝑎
Using division rule of Log 𝑙𝑜𝑔10 𝑏 = 𝑙𝑜𝑔10 𝑎 − 𝑙𝑜𝑔10 𝑏
𝐴 = 𝑙𝑜𝑔10
100
= 𝑙𝑜𝑔10 100 − 𝑙𝑜𝑔10 %𝑇
%𝑇
(7)
So,
𝐴 = 𝑙𝑜𝑔10 100 − 𝑙𝑜𝑔10 %𝑇
(8)
𝐴 = 𝑙𝑜𝑔10 102 − 𝑙𝑜𝑔10 %𝑇
(9)
𝐴 = 2𝑙𝑜𝑔10 10 − 𝑙𝑜𝑔10 %𝑇
𝐴 = 2 − 𝑙𝑜𝑔10 %𝑇
∴ 𝑙𝑜𝑔10 𝑎2 = 2𝑙𝑜𝑔10 𝑎
(10)
∴ 𝑙𝑜𝑔10 10 = 1
(11)
𝐴 = 2 − 𝑙𝑜𝑔10 %𝑇
(12)
The last equation, A = 2 - log10 %T , is worth remembering because it allows you to easily calculate
absorbance from percentage transmittance data.
The value of transmitted light %𝑇 varies from 0 to 100.
Case 1: When no light pass through sample i.e. %𝑇 = 0,
𝐴 = 2 − 𝑙𝑜𝑔10 0 = 2 − 0 = 2
(13)
Case 2: When all incident light passes through sample i.e. %𝑇 = 100,
𝐴 = 2 − 𝑙𝑜𝑔10 100 = 2 − 𝑙𝑜𝑔10 102 = 2 − 2𝑙𝑜𝑔10 10 = 2 − 2 = 0
(14)
It is clear from both cases that the maximum value of Absorption, 𝐴 = 2.
Hence proved that the absorption value of spectrometer cannot be greater than 2.
𝑨𝒃𝒔𝒐𝒓𝒑𝒕𝒊𝒐𝒏, 𝑨 ≤ 𝟐
The relationship between absorbance and transmittance is illustrated in the following diagram
which shows that the value of Absorbance is less than or equal to 2 for all values of transmittance: