Certainly, let's walk through the solutions step-by-step.
### 3. What absorption value corresponds to 45% light transmittance?
The relationship between transmittance (\(\( T \)\)) and absorption (\(\( A \)\)) can be defined using the formula:
\(\[ A = -\log_{10}(T) \]\)
Here, transmittance (\(\( T \)\)) is given as a percentage, so we first need to convert it to a decimal form by dividing by 100.
1. Given transmittance \(\( T \)\) = 45%, converting this to a decimal:
\(\[ T = \frac{45}{100} = 0.45 \]\)
2. Now, using the formula:
\(\[ A = -\log_{10}(0.45) \]\)
3. To find \(\( A \)\), you would take the logarithm base 10 of 0.45 (which is approximately -0.3468) and then multiply by -1:
\(\[ A \approx 0.347 \]\)
So, the absorption value corresponding to 45% light transmittance is approximately 0.347.
### 4. If the transmittance of a solution with a concentration of 0.0100 mol/L is 45.0%, what will be the transmittance if the concentration is 0.0200 mol/L?
To solve this problem, we will use the Beer-Lambert law which states that absorption (\(\( A \)\)) is directly proportional to the concentration (\(\( c \)\)) of the solution. The proportional relationship is given by:
\(\[ A = \epsilon \cdot c \cdot l \]\)
where:
- \(\( \epsilon \)\) is the molar absorptivity (a constant),
- \(\( c \)\) is the concentration,
- \(\( l \)\) is the path length (also a constant).
From this law, if we double the concentration, the absorption should also double, given that the molar absorptivity and path length remain constant.
1. Calculate the absorption value (\(\( A \)\)) for the 45% transmittance.
- Given earlier: \(\( A = -\log_{10}(0.45) \)\)
- We calculated this to be approximately \(\( A = 0.347 \)\).
2. Now, double the concentration from 0.0100 mol/L to 0.0200 mol/L.
- If the concentration is doubled, the new absorption (\(\( A' \)\)) will be:
\(\[ A' = 2 \times 0.347 = 0.694 \]\)
3. To find the new transmittance corresponding to this new absorption value, we reverse the earlier formula to solve for \(\( T \)\):
\(\[ T' = 10^{-A'} \]\)
Substitute \(\( A' \)\):
\(\[ T' = 10^{-0.694} \]\)
4. Calculate this using a calculator:
\(\[ T' \approx 0.202 \]\)
Convert this to a percentage:
\(\[ T' \approx 20.2\% \]\)
So, the transmittance of the solution at a concentration of 0.0200 mol/L will be approximately 20.2%.
