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We are provided with a scenario where the absorbances of a solution containing drugs X and Y are measured at two different wavelengths (269 nm and 315 nm), and their molar absorptivities at these wavelengths are given. Using Beer-Lambert's Law, we need to determine the concentrations of X and Y by constructing and solving a system of linear equations.

### Beer-Lambert's Law:
\[ A = \varepsilon \times c \times l \]
Where:
- \( A \) is the absorbance.
- \( \varepsilon \) is the molar absorptivity (dm³ mol⁻¹ cm⁻¹).
- \( c \) is the concentration (mol dm⁻³).
- \( l \) is the path length of the quartz cell (cm), which we can assume to be the same for both measurements.

### Information Given:
- Absorbance at 269 nm: \( A_{269} = 0.948 \)
- Absorbance at 315 nm: \( A_{315} = 0.562 \)
- Molar absorptivities of X and Y:
    - \( \varepsilon_{X,269} = 16400 \), \( \varepsilon_{X,315} = 3990 \)
    - \( \varepsilon_{Y,269} = 3870 \), \( \varepsilon_{Y,315} = 6420 \)

### Step-by-Step Setup:

We have two wavelengths and two unknown concentrations, \( c_X \) and \( c_Y \), for drugs X and Y, respectively. So, we can form two linear equations from the absorbances at 269 nm and 315 nm.

#### Equation 1 (for 269 nm):
\[
A_{269} = \varepsilon_{X,269} \times c_X \times l + \varepsilon_{Y,269} \times c_Y \times l
\]
Substituting the values:
\[
0.948 = 16400 \times c_X \times l + 3870 \times c_Y \times l
\]

#### Equation 2 (for 315 nm):
\[
A_{315} = \varepsilon_{X,315} \times c_X \times l + \varepsilon_{Y,315} \times c_Y \times l
\]
Substituting the values:
\[
0.562 = 3990 \times c_X \times l + 6420 \times c_Y \times l
\]

### Simplified System of Equations (assuming \( l = 1 \) cm):
\[
0.948 = 16400 \times c_X + 3870 \times c_Y \quad \text{(1)}
\]
\[
0.562 = 3990 \times c_X + 6420 \times c_Y \quad \text{(2)}
\]

### Solving the System of Equations:

Let me solve these equations to find the concentrations of \( c_X \) and \( c_Y \). I will use matrix algebra to do this.The concentrations of drugs X and Y in the solution are:

- Concentration of X (\(c_X\)): \(4.35 \times 10^{-5} \, \text{mol/dm}^3\)
- Concentration of Y (\(c_Y\)): \(6.05 \times 10^{-5} \, \text{mol/dm}^3\)

Would you like further details or have any additional questions?

Here are 5 related questions you might explore:
1. How would changes in the path length \( l \) affect the concentrations?
2. How would the solution change if absorbance measurements were slightly off?
3. Can the Beer-Lambert law be applied if the solution is non-homogeneous?
4. What happens if the molar absorptivity values are similar for both substances?
5. How would the concentrations change if the absorbances at different wavelengths were swapped?

**Tip**: Always ensure that the units of absorptivity, concentration, and path length are consistent when applying Beer-Lambert’s law.

A mixture of drugs X and Y from a capsule is dissolved in methanol and made up to a known volume. X and Y have peaks at 269nm and 315nm respectively in the ultraviolet spectrum. The absorbance of the solution of the mixture was measured to be 0.948 at the wavelength 269 nm and 0.562 at the wavelength 315 nm. Construct a system of linear equations that will allow you to determine the concentrations of X and Y, using matrix algebra.

This guide demonstrates how to use Beer-Lambert's law and matrix algebra to determine the concentrations of drugs X and Y from absorbance measurements at 269nm and 315nm. Suitable for undergraduate students in chemistry or physics.

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