Problem 14 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Four solutions of a dye were prepared in water. In a \(1.00 \mathrm{cm}\) cell, the percentage of light at a particular wavelength transmitted through each solution is given in the following table. $$\begin{array}{lllll}\hline \text { Concentration/moldm }^{3} & 0.004 & 0.010 & 0.020 & 0.040 \\\\\text { Percentage light transmitted } &79 & 56 & 32 & 10 \\\\\hline\end{array}$$ Confirm that these data obey the Beer-Lambert law and calculate the molar absorption coefficient, \(\varepsilon,\) at this wavelength. (Section \(11.4)\)

Step-by-Step Solution

Answer

Yes, the data obeys Beer-Lambert law. The molar absorption coefficient \(\varepsilon\) is constant across the concentrations.

Step 1: Understand Beer-Lambert Law

The Beer-Lambert law is given by the equation \(A = e\times c\times l\), where \(A\) is the absorbance, \(\varepsilon\) is the molar absorptivity or absorption coefficient, \(c\) is the concentration, and \(l\) is the path length of the cell. Since the light path length \(l\) is \(1.00 \text{ cm}\) in this case, it simplifies the calculation.

Step 2: Convert Percentage Transmittance to Absorbance

The relationship between absorbance \(A\) and percentage transmittance \(T\) is given by \(A = -\log_{10}(\frac{T}{100})\). We need to convert the given percentage transmittances into absorbance values for each concentration.

Step 3: Calculate Absorbance for Each Concentration

Calculate absorbance for each concentration from the percentage transmittances:- For \(c = 0.004\, \text{mol}\, \text{dm}^{-3}\), \(T = 79\%\), \(A = -\log_{10}(0.79)\).- For \(c = 0.010\, \text{mol}\, \text{dm}^{-3}\), \(T = 56\%\), \(A = -\log_{10}(0.56)\).- For \(c = 0.020\, \text{mol}\, \text{dm}^{-3}\), \(T = 32\%\), \(A = -\log_{10}(0.32)\).- For \(c = 0.040\, \text{mol}\, \text{dm}^{-3}\), \(T = 10\%\), \(A = -\log_{10}(0.10)\).

Step 4: Confirm Beer-Lambert Law

For the Beer-Lambert law to hold, the absorbance \(A\) should be directly proportional to the concentration \(c\). Calculate and compare \(A/c\) for each concentration to check if \e remains constant.

Step 5: Calculate Molar Absorption Coefficient \(\varepsilon\)

Given the absorbance \(A\) and calculated that \(A/c\) is constant, determine the molar absorption coefficient, \(\varepsilon\). Since it should be the same for each pair of \(A\) and \(c\), use any concentration to determine \(\varepsilon = \frac{A}{c} \cdot l\).

Key Concepts

Absorbance

Absorbance is a measure of the amount of light absorbed by a solution and is a key concept in understanding how the Beer-Lambert Law operates. When light passes through a solution, some of it can be absorbed by the molecules within that solution.
The absorbance, denoted as \( A \), can be calculated through a mathematical relationship with transmittance (the proportion of light that passes through the solution). This relationship is described by the formula:

• \( A = -\log_{10}(\frac{T}{100}) \)

Here, \( T \) is the percentage transmittance of light, ranging from 0% to 100%. Transforming transmittance into absorbance allows us to connect it with the concentration of the solution.
A higher absorbance value indicates greater light absorption, providing insights into the solution's concentration and its interaction with light.

Molar absorption coefficient

The molar absorption coefficient, represented by the Greek letter \( \varepsilon \), is a crucial parameter in the Beer-Lambert Law. It reflects how well a substance absorbs light at a particular wavelength. Think of it as a fingerprint for each molecule, unique to its interaction with light.
In the context of Beer-Lambert law, the absorbance \( A \) of a solution is proportional to the concentration \( c \) of the absorbing species and the path length \( l \), measured usually in centimeters:

• \( A = \varepsilon \times c \times l \)

By rearranging this equation, \( \varepsilon \) can be determined as:

• \( \varepsilon = \frac{A}{c \times l} \)

This coefficient is particularly useful because it allows scientists to compare the light-absorbing properties of different substances. Furthermore, since \( \varepsilon \) is consistent for a given substance at a specific wavelength, it simplifies the calculation of concentrations by providing a direct relation to absorbance.

Transmittance to Absorbance Conversion

Converting transmittance to absorbance is essential for analysis using the Beer-Lambert law because it provides a linear relationship between absorbance and concentration. Transmittance \( T \) is expressed as a percentage of incident light that a solution permits to pass through it. However, on its own, it isn't directly proportional to concentration.
The formula to convert transmittance to absorbance is:

• \( A = -\log_{10}(\frac{T}{100}) \)

This logarithmic relationship linearizes the data from transmittance measurements, making it easier to analyze the impact of concentration changes. A transmittance of 79% (where 79% of light passes through a solution) can be converted to absorbance by calculating \(-\log_{10}(0.79)\).
This conversion is fundamental to confirming the Beer-Lambert law in laboratory settings, as it allows for easy checking of proportionality between absorbance and concentration.