Problem 3 - Analytical Chemistry

Question

A solution's transmittance is \(35.0 \%\). What is the transmittance if you dilute \(25.0 \mathrm{~mL}\) of the solution to \(50.0 \mathrm{~mL}\) ?

Step-by-Step Solution

Answer

The new transmittance is 59.4%.

Step 1: Understand the Problem

We have a solution with an initial transmittance of 35.0%. It is diluted by doubling its volume from 25.0 mL to 50.0 mL. We need to find the new transmittance of this diluted solution.

Step 2: Recall the Beer-Lambert Law

The Beer-Lambert Law relates absorbance (A) to concentration (c) and has the formula: \( A = \epsilon \cdot c \cdot l \), where \( \epsilon \) is the molar absorptivity, \( c \) is the concentration, and \( l \) is the path length.

Step 3: Determine the Relationship Between Transmittance and Absorbance

Transmittance (T) is related to absorbance (A) by the formula: \( T = 10^{-A} \), or equivalently \( A = -\log_{10}(T) \). For a 35.0% transmittance, \( T = 0.35 \).

Step 4: Calculate the Initial Absorbance

Using the relation \( A = -\log_{10}(T) \), the initial absorbance \( A_1 \) is calculated as: \[ A_1 = -\log_{10}(0.35) = 0.456 \]

Step 5: Understand the Effect of Dilution on Concentration and Absorbance

Diluting the solution decreases its concentration proportionally. Since the volume is doubled (25.0 mL to 50.0 mL), the concentration is halved. The new absorbance \( A_2 \) is half of \( A_1 \): \( A_2 = \frac{A_1}{2} = 0.228 \).

Step 6: Calculate the New Transmittance

Now that we have \( A_2 = 0.228 \), calculate the new transmittance \( T_2 \) using \( T_2 = 10^{-A_2} \): \[ T_2 = 10^{-0.228} = 0.594 \] Expressing 0.594 as a percentage gives a new transmittance of 59.4%.

Key Concepts

Transmittance

Transmittance refers to the measure of the fraction of light that passes through a sample. It is expressed as a percentage and signifies how much light is not absorbed by the solution. For a transmittance of 35.0%, this means 35% of the light directed at the solution is passing through, while the remaining 65% is absorbed. This concept is crucial when studying light interactions with materials, as it provides insights into the concentration of substances within the solution. To calculate transmittance, the formula used is typically based on the absorbance value, where transmittance T is equals to:

• \( T = 10^{-A} \)

Understanding this relationship is vital for determining the properties and behavior of different solutions when they are exposed to light.

Absorbance

Absorbance is a measure that indicates how much light a sample absorbs at a particular wavelength, and it is directly related to the concentration of the substance within the sample. The formula linking absorbance and transmittance is given by:

• \( A = -\log_{10}(T) \)

where A is absorbance, and T is transmittance. An important aspect of absorbance is its logarithmic nature, meaning small changes in absorbance can result in significant alterations in transmittance. For an initial solution with 35.0% transmittance, the calculated absorbance would be approximately 0.456.
Absorbance helps us understand the Beer-Lambert Law's role in determining concentrations. When you understand that a higher absorbance means more light is absorbed by the sample, you start seeing the direct relationship between light behavior and sample characteristics.

Dilution Effect

Dilution occurs when you add a solvent to a solution, thus decreasing the concentration of the solute. Practically, this is seen when a solution is made less concentrated by increasing its volume. In the context of the Beer-Lambert Law, diluting a solution affects its absorbance and transmittance.
When a solution is diluted, as in the given exercise where the volume is doubled from 25.0 mL to 50.0 mL, the concentration decreases by half. Consequently, the absorbance is halved, and the new absorbance can be calculated using the relationship:

• New absorbance: \( A_2 = \frac{A_1}{2} \)

This understanding allows us to predict changes in optical properties of the solution. The new transmittance, calculated from the lowered absorbance, increases to indicate more light passes through the solution post-dilution.

Concentration

Concentration describes the amount of solute per unit volume of solution and is a fundamental concept in chemistry. It directly affects the solution's optical properties, as described by the Beer-Lambert Law.
The concentration determines how much light a solution absorbs, and thus influences both the absorbance and transmittance values. In our example, by diluting the solution from 25.0 mL to 50.0 mL, the concentration of the solute is halved. This results in a halved absorbance, computing using the formula, as the number of absorbing molecules in the path of light has decreased by half.
Understanding this concept is crucial when using spectrophotometric methods to determine concentrations, as it provides a basis for correlating measured absorbance with actual solute concentrations in a detailed manner, guiding practical applications in laboratory settings.