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In the given exercise, you are tasked with determining the concentrations of two salts (A and B) in an unknown sample by calculating the molar absorption coefficients at two wavelengths (510 nm and 575 nm). You will then use these coefficients to find the molar concentrations of A and B in the unknown solution.

### Part 1: Calculate the 4 Molar Absorption Coefficients

The absorbance equation is given by **Beer-Lambert Law**:

\[
A = \varepsilon \cdot c \cdot l
\]

Where:
- \( A \) is the absorbance,
- \( \varepsilon \) is the molar absorption coefficient (to be calculated),
- \( c \) is the concentration (provided),
- \( l \) is the path length (1 cm as stated).

We can rearrange this equation to solve for \( \varepsilon \):

\[
\varepsilon = \frac{A}{c \cdot l}
\]

The data given are:
- Concentration of A = 0.15 M,
- Concentration of B = 0.06 M,
- Path length \( l = 1 \) cm.

### Absorbance data for:
- **510 nm**:
  - \( A_A = 0.714 \), \( A_B = 0.298 \)
- **575 nm**:
  - \( A_A = 0.0097 \), \( A_B = 0.757 \)

We will now calculate the molar absorption coefficients for each case:

1. **\( \varepsilon_A(510) \)**: for compound A at 510 nm.
2. **\( \varepsilon_A(575) \)**: for compound A at 575 nm.
3. **\( \varepsilon_B(510) \)**: for compound B at 510 nm.
4. **\( \varepsilon_B(575) \)**: for compound B at 575 nm.

Let's calculate these coefficients.The molar absorption coefficients are:

1. **\( \varepsilon_A(510) \)** = 4.76 L·mol⁻¹·cm⁻¹ (for compound A at 510 nm)
2. **\( \varepsilon_A(575) \)** = 0.065 L·mol⁻¹·cm⁻¹ (for compound A at 575 nm)
3. **\( \varepsilon_B(510) \)** = 4.97 L·mol⁻¹·cm⁻¹ (for compound B at 510 nm)
4. **\( \varepsilon_B(575) \)** = 12.62 L·mol⁻¹·cm⁻¹ (for compound B at 575 nm)

### Part 2: Calculate the Concentrations of A and B in the Solution

The absorbances for the unknown solution at 510 nm and 575 nm are:
- At 510 nm: \( A = 0.4 \)
- At 575 nm: \( A = 0.577 \)

The system of equations for the absorbance values at both wavelengths is:

\[
A_{510} = \varepsilon_A(510) \cdot c_A + \varepsilon_B(510) \cdot c_B
\]
\[
A_{575} = \varepsilon_A(575) \cdot c_A + \varepsilon_B(575) \cdot c_B
\]

We will solve these equations to find the concentrations \( c_A \) and \( c_B \). Let's calculate these values next.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, you can take the next step manually. Here's how to solve the system of equations for the concentrations \( c_A \) and \( c_B \):

1. Use the equations:
   \[
   0.4 = 4.76 \cdot c_A + 4.97 \cdot c_B
   \]
   \[
   0.577 = 0.065 \cdot c_A + 12.62 \cdot c_B
   \]

2. Solve this system of linear equations for \( c_A \) and \( c_B \), which are the concentrations of compounds A and B in the sample.

Do you have any questions, or would you like more details on how to solve these equations by hand?

Here are 5 related questions to consider:
1. How does Beer-Lambert law apply to multi-component solutions?
2. What happens to absorbance if the path length changes in the cuvette?
3. Can molar absorption coefficients be negative? Why or why not?
4. How do you interpret the relationship between concentration and absorbance?
5. How would the result change if the concentrations of A and B were swapped?

**Tip:** Always double-check units in calculations to avoid errors, especially in multi-step problems.

On veut déterminer la concentration de deux sels A et B dans un échantillon inconnu en solution aqueuse. On enregistre un spectre dans le visible de chacun de ces deux composés pris séparément en solution aqueuse, ainsi que la solution échantillon à analyser. Le trajet optique des cuves utilisées est de 1 cm. Les valeurs des absorbances mesurées à 510 et 575 nm sur les trois spectres sont les suivantes:

Composé A (C = 0,15M)
- 510 nm: A = 0,714
- 575 nm: A = 0,0097

Composé B (C = 0,06M)
- 510 nm: A = 0,298
- 575 nm: A = 0,757

Solution échantillon
- 510 nm: A = 0,4
- 575 nm: A = 0,577

Calculer les coefficients d'absorption molaires εA(510), εA(575), εB(510), εB(575). Calculer ensuite les concentrations molaires de A et de B dans la solution échantillon.

Learn how to calculate the molar absorption coefficients for two compounds at different wavelengths using Beer-Lambert Law. Detailed step-by-step process for finding εA and εB at 510 nm and 575 nm, followed by calculating the concentrations of A and B in an unknown sample.

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