![23. A 0.1% (w/v) solution of a protein absorbs 20% of the incident light. What fraction of light is transmitted if the concentration is increased to 0.4%? [Correct to two decimal places]](https://www.letstalkacademy.com/wp-content/uploads/2026/07/01-24.jpg)
23. A 0.1% (w/v) solution of a protein absorbs 20% of the incident light. What fraction of light is transmitted if the concentration is increased to 0.4%? [Correct to two decimal places]
Fraction of Light Transmitted When Protein Concentration Increases from 0.1% to 0.4%
Correct Answer
Correct Answer: 0.41
A 0.1% (w/v) protein solution absorbs 20% of the incident light, which means that it transmits the remaining 80% of the light. Therefore, the initial fractional transmittance is 0.80.
When the protein concentration increases from 0.1% to 0.4%, the concentration becomes four times greater. According to the Beer-Lambert law, absorbance, rather than percentage absorption, is directly proportional to concentration. Therefore, the initial transmittance must first be related to absorbance. After accounting for the fourfold increase in concentration, the new transmittance is found to be 0.4096, which is 0.41 when rounded to two decimal places.
Therefore:
Fraction of light transmitted = 0.41
Understanding the Information Given in the Question
A 20% Absorption Means 80% Transmission
The question states that the original protein solution absorbs 20% of the incident light. If all the incident light is represented as 100%, the portion that is not absorbed is transmitted through the solution.
Percentage of light transmitted = 100% − 20%
Percentage of light transmitted = 80%
To express this value as fractional transmittance:
T1 = 80/100
Therefore:
T1 = 0.80
Thus, the 0.1% protein solution transmits 0.80 or 80% of the incident light.
Why We Cannot Simply Multiply 20% Absorption by Four
Absorbance and Percentage Absorption Are Different Quantities
The protein concentration increases from:
0.1% to 0.4%
The concentration has therefore increased by a factor of:
0.4/0.1 = 4
It may appear tempting to multiply the original 20% absorption by four and conclude that the new solution absorbs 80% and therefore transmits 20%. However, this approach is incorrect.
The Beer-Lambert law states that absorbance is directly proportional to concentration. It does not state that the percentage of light absorbed is directly proportional to concentration.
The correct relationship is:
A = εcl
where A is absorbance, ε is the molar absorptivity or extinction coefficient, c is concentration and l is the optical path length.
Since the concentration becomes four times greater while the protein, wavelength and path length remain unchanged:
A2 = 4A1
This relationship must be used to calculate the new fraction of transmitted light.
Step-by-Step Calculation Using the Beer-Lambert Law
Step 1: Calculate the Initial Fractional Transmittance
The original solution absorbs 20% of the incident light. Therefore, it transmits:
100% − 20% = 80%
Hence:
T1 = 0.80
Step 2: Convert the Initial Transmittance into Absorbance
Absorbance and transmittance are related by:
A = −log10T
Therefore:
A1 = −log10(0.80)
Calculating the logarithm:
A1 = 0.09691
Thus, the absorbance of the 0.1% protein solution is approximately:
A1 = 0.0969
Step 3: Determine the Increase in Concentration
The initial concentration is:
c1 = 0.1%
The final concentration is:
c2 = 0.4%
Therefore:
c2/c1 = 0.4/0.1 = 4
The final solution is four times as concentrated as the initial solution.
According to the Beer-Lambert law:
A ∝ c
Therefore:
A2/A1 = c2/c1
Hence:
A2 = 4A1
Substituting the initial absorbance:
A2 = 4 × 0.09691
Therefore:
A2 = 0.38764
Step 4: Convert the New Absorbance Back into Transmittance
The relationship between absorbance and transmittance is:
A = −log10T
Therefore:
T = 10−A
For the new protein concentration:
T2 = 10−0.38764
This gives:
T2 = 0.4096
Correct to two decimal places:
T2 = 0.41
Therefore, the fraction of incident light transmitted by the 0.4% protein solution is:
0.41
A Faster Mathematical Method
Using the Relationship Between Concentration and Transmittance
The same calculation can be solved more directly. Since:
A = −log10T
and absorbance is proportional to concentration:
A2 = 4A1
Therefore:
−log10T2 = 4(−log10T1)
Hence:
log10T2 = 4log10T1
Using the logarithmic rule:
log(xn) = n log(x)
we obtain:
T2 = (T1)4
Since:
T1 = 0.80
therefore:
T2 = (0.80)4
T2 = 0.80 × 0.80 × 0.80 × 0.80
T2 = 0.4096
Therefore, correct to two decimal places:
T2 = 0.41
This direct method is especially useful when the concentration changes by an exact multiple.
Why the Answer Is 0.41 and Not 0.20
Beer-Lambert Law Applies to Absorbance
An incorrect approach would be to assume that because the concentration increases fourfold, the percentage absorption must also increase fourfold:
Original absorption = 20%
New absorption = 4 × 20% = 80%
New transmission = 20% = 0.20
This reasoning does not follow the Beer-Lambert law because it assumes that the percentage of absorbed light is directly proportional to concentration.
In reality:
Absorbance ∝ Concentration
but percentage absorption is not directly proportional to concentration.
The relationship between absorbance and transmittance is logarithmic:
A = −log10T
Therefore, when the concentration increases fourfold, the absorbance increases fourfold, but the percentage of absorbed light does not simply increase fourfold.
The correct transmitted fraction is therefore 0.41, not 0.20.
Understanding Absorbance, Transmittance and Percentage Transmission
Fractional Transmittance
Fractional transmittance is defined as:
T = I/I0
where I0 is the intensity of incident light and I is the intensity of transmitted light.
A transmittance of T = 0.80 means that 80% of the original light passes through the sample. Similarly, T = 0.41 means that approximately 41% of the incident light passes through the sample.
Percentage Transmittance
Percentage transmittance is calculated as:
%T = T × 100
Therefore, for the final solution:
%T = 0.4096 × 100
%T = 40.96%
Thus, the final solution transmits approximately 40.96% of the incident light. However, the question asks for the fraction of light transmitted, so the required answer is 0.41.
Absorbance
Absorbance measures how strongly a sample attenuates light and is related logarithmically to transmittance:
A = −log10T
As transmittance decreases, absorbance increases. Therefore, a more concentrated protein solution generally has greater absorbance and lower transmittance.
Physical Interpretation of the Result
Increasing Protein Concentration Reduces Light Transmission
At a protein concentration of 0.1%, the sample transmits 80% of the incident light:
T1 = 0.80
When the concentration increases to 0.4%, there are more light-absorbing molecules in the path of the incident beam. Consequently, a greater fraction of photons is absorbed, and less light reaches the detector.
The transmitted fraction decreases to:
T2 = 0.4096 ≈ 0.41
Thus, increasing the protein concentration fourfold reduces the transmitted light from 80% to approximately 41%.
General Formula for Similar Questions
When Concentration Changes by a Factor of n
Suppose the initial transmittance is T1, and the concentration increases by a factor of n.
Since absorbance is directly proportional to concentration:
A2 = nA1
Using:
A = −log10T
we obtain:
T2 = (T1)n
Therefore, the general relationship is:
T2 = (T1)(c2/c1)
For the present question:
T1 = 0.80
and:
c2/c1 = 0.4/0.1 = 4
Therefore:
T2 = (0.80)4
T2 = 0.4096
T2 ≈ 0.41
Complete Calculation at a Glance
| Quantity | Value |
|---|---|
| Initial protein concentration | 0.1% (w/v) |
| Light absorbed initially | 20% |
| Light transmitted initially | 80% |
| Initial fractional transmittance | 0.80 |
| Final protein concentration | 0.4% (w/v) |
| Increase in concentration | 4-fold |
| Final transmittance | (0.80)4 |
| Calculated value | 0.4096 |
| Answer to two decimal places | 0.41 |
Final Answer
A 0.1% protein solution absorbs 20% of the incident light, so its initial fractional transmittance is:
T1 = 1 − 0.20 = 0.80
The concentration increases from 0.1% to 0.4%, which is a fourfold increase. According to the Beer-Lambert law, absorbance is directly proportional to concentration. Therefore:
T2 = (T1)4
T2 = (0.80)4
T2 = 0.4096
Correct to two decimal places:
Correct Answer: 0.41
