Introduction:
In this problem, we are given the absorbance of a protein solution at a certain concentration and we need to determine the fraction of light transmitted when the concentration is increased. The concentration of the protein solution is given as a percentage (w/v) and the absorbance is given as a percentage of the incident light.

Given:
- Concentration of protein solution: 0.1% (w/v)
- Absorbance of the protein solution: 20% of the incident light

Calculating the Absorption Coefficient:
To find the fraction of light transmitted, we need to first calculate the absorption coefficient of the protein solution. The absorption coefficient is a measure of how strongly a substance absorbs light at a specific wavelength.

The absorption coefficient (ε) can be calculated using the Beer-Lambert law, which states that the absorbance (A) of a solution is directly proportional to the concentration (c) of the absorbing species and the path length (l) of the light through the solution. Mathematically, this can be represented as:

A = εcl

Given that the absorbance is 20% (or 0.2) and the concentration is 0.1% (or 0.001), we can rearrange the equation to solve for ε:

ε = A / (cl) = 0.2 / (0.001 * l) = 200 / l

Calculating the Transmittance:
The transmittance (T) is the fraction of light transmitted through a solution and is calculated as the ratio of the transmitted light intensity to the incident light intensity. It can be represented as:

T = I_t / I_0

where I_t is the transmitted light intensity and I_0 is the incident light intensity.

Since absorbance is related to transmittance by the equation:

A = -log(T)

we can rearrange this equation to solve for T:

T = 10^(-A)

Increasing the Concentration:
Now, let's calculate the fraction of light transmitted when the concentration of the protein solution is increased to 0.4% (or 0.004).

Using the same absorption coefficient (ε) calculated earlier, we can substitute it into the Beer-Lambert law equation with the new concentration (0.004) and the same path length (l):

A' = ε * c' * l = (200 / l) * 0.004 * l = 0.8

Substituting this new absorbance value into the transmittance equation:

T' = 10^(-A') = 10^(-0.8) ≈ 0.158

Therefore, when the concentration is increased to 0.4% (w/v), approximately 15.8% of the incident light is transmitted.

Conclusion:
In conclusion, we have determined that when the concentration of the protein solution is increased from 0.1% to 0.4%, the fraction of light transmitted is approximately 15.8%. This calculation was based on the Beer-Lambert law and the relationship between absorbance and transmittance.