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