Problem 8 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

\(\mathrm{A} 5.0 \times 10^{-4} \mathrm{mol} \mathrm{dm}^{-3}\) solution of \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) absorbed \(64 \%\) of the incident light when placed in a \(2.0 \mathrm{cm}\) cell at a wavelength where \(\mathrm{CCl}_{4}\) does not absorb. Calculate the molar absorption coefficient of \(\mathrm{Br}_{2}\). (Section 10.3 )

Step-by-Step Solution

Answer

The molar absorption coefficient is \( 444 \text{ dm}^3 \text{ mol}^{-1} \text{ cm}^{-1} \).

Step 1: Understand the Beer-Lambert Law

The Beer-Lambert Law relates the absorption of light to the properties of the material through which the light is traveling. It is given by the equation: \( A = \varepsilon c l \), where \( A \) is the absorbance, \( \varepsilon \) is the molar absorption coefficient, \( c \) is the concentration of the solution, and \( l \) is the path length of the cell the light passes through. In this exercise, we need to find \( \varepsilon \). We are given \( c = 5.0 \times 10^{-4} \text{ mol dm}^{-3} \), \( l = 2.0 \text{ cm} \), and if the solution absorbs 64% of the light, the transmittance (T) is 0.36.

Step 2: Calculate Absorbance

The absorbance \( A \) can be calculated from the transmittance using the formula: \( A = -\log_{10}(T) \).Given that the solution absorbs 64% of the light, the transmittance \( T \) is 0.36 (since 100% - 64% = 36%).Let's calculate:\[A = -\log_{10}(0.36)\]

Step 3: Compute Absorbance Numerically

Let's compute the absorbance:\[A = -\log_{10}(0.36) \approx 0.444\]

Step 4: Rearrange Beer-Lambert Formula

To find the molar absorption coefficient \( \varepsilon \), rearrange the Beer-Lambert Law formula:\[\varepsilon = \frac{A}{c \cdot l}\]With \( A = 0.444 \), \( c = 5.0 \times 10^{-4} \text{ mol dm}^{-3} \), and \( l = 2.0 \text{ cm} \), substitute these values into the formula.

Step 5: Calculate Molar Absorption Coefficient

Let's substitute the known values:\[\varepsilon = \frac{0.444}{5.0 \times 10^{-4} \times 2.0}\]Perform the calculation:\[\varepsilon = \frac{0.444}{1.0 \times 10^{-3}} = 444 \text{ dm}^3 \text{ mol}^{-1} \text{ cm}^{-1}\]

Step 6: Compile Results

After calculating, the molar absorption coefficient of \( \mathrm{Br}_{2} \) is found to be \( 444 \text{ dm}^3 \text{ mol}^{-1} \text{ cm}^{-1} \). This coefficient represents the efficiency with which \( \mathrm{Br}_{2} \) in the cell absorbs light at the given wavelength.

Key Concepts

Beer-Lambert Law

The Beer-Lambert Law is a fundamental principle in chemistry that helps us understand how light interacts with materials. It describes the linear relationship between absorbance and both the concentration of the absorbing species and the path length of light. The law's formula is given by:\[ A = \varepsilon c l \]Where:

• \( A \) is the absorbance of the solution, a dimensionless number that represents the amount of light absorbed.
• \( \varepsilon \) is the molar absorption coefficient, a measure of how strongly a chemical absorbs light at a particular wavelength.
• \( c \) is the concentration of the molar substance in mol \( \text{dm}^{-3} \).
• \( l \) is the path length that light travels through, usually measured in centimeters.

This law is crucial because it allows us to pinpoint the relationship between these different variables. It implies that if we know three of the variables, we can find the fourth. This exercise revolves around calculating \( \varepsilon \) using the known concentration, path length, and absorbance. Understanding this relationship is key to performing any calculation involving absorbance.

Absorbance Calculation

To find absorbance, we use the relation between absorbance and transmittance in solutions. Transmittance \( T \) is the ratio of the intensity of light passing through the sample to the light intensity before it enters the sample. Absorbance \( A \) can be calculated from transmittance using the formula:\[ A = -\log_{10}(T) \]When given that a sample absorbs 64% of the light, we deduce that only 36% of light is transmitted (because 100% - 64% = 36%). This means the transmittance \( T \) is 0.36. We plug this value into the formula:\[ A = -\log_{10}(0.36) \approx 0.444 \]This result signifies the absorbance by calculating how much of the light does not penetrate through the sample. By understanding absorbance, we can further deduce the concentration and chemical behavior of substances in solutions. This knowledge is fundamental in being able to apply the Beer-Lambert Law effectively.

Transmittance

Transmittance is a measure that describes the fraction of light that successfully passes through a material. In simpler terms, it measures how much light "gets through" a solution. Mathematically, it is expressed as:\[ T = \frac{I}{I_0} \]Where:

• \( I \) is the intensity of light after passing through the sample.
• \( I_0 \) is the intensity of light before it enters the sample.

In scenarios where we know the percentage of light absorbed, we can easily determine transmittance by subtracting the percentage absorbed from 100%. For example, if 64% of the light is absorbed, then 36% is transmitted, leading to a transmittance value of 0.36.Transmittance values range from 0 (complete absorption) to 1 (no absorption). Understanding transmittance helps in deriving absorbance, pivotal in quantitative analysis of solutions. This concept ties directly into assessments of concentrations and chemical interactions in solutions, complementing the foundational principles of the Beer-Lambert Law.