Problem 78 - Algebra and Trigonometry

Question

Absorption of Light A spectrophotometer measures the concentration of a sample dissolved in water by shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light absorbed, we can calculate the concentration of the sample. For a certain substance, the concentration (in moles/liter) is found using the formula $$C=-2500 \ln \left(\frac{I}{I_{0}}\right)$$ where \(I_{0}\) is the intensity of the incident light and \(I\) is the intensity of light that emerges. Find the concentration of the substance if the intensity \(I\) is 70\(\%\) of \(I_{0} .\)

Step-by-Step Solution

Answer

The concentration is approximately 891.75 moles/liter.

Step 1: Understand the given values

The problem states that the intensity of the light that emerges (\(I\)) is 70\(\%\) of the incident light's intensity (\(I_0\)). This means that \(I = 0.7 \times I_0\).

Step 2: Apply the formula for concentration

Use the formula given: \[ C = -2500 \ln \left( \frac{I}{I_0} \right) \]Substitute \(I = 0.7I_0\) into the equation, so it becomes:\[ C = -2500 \ln \left( \frac{0.7I_0}{I_0} \right) \]

Step 3: Simplify the fraction

The fraction \(\frac{0.7I_0}{I_0}\) simplifies to \(0.7\). Therefore, the formula becomes:\[ C = -2500 \ln(0.7) \]

Step 4: Calculate the natural logarithm

Use a calculator to find the value of \(\ln(0.7)\). The natural logarithm of 0.7 is approximately \(-0.3567\).

Step 5: Calculate the concentration

Substitute \(\ln(0.7) \approx -0.3567\) into the equation:\[ C = -2500 \times -0.3567 \]Calculate this product to find the concentration \(C\). The concentration is approximately \(891.75\) moles/liter.

Key Concepts

Absorption of Light

When we say a substance absorbs light, it means that not all light passing through the substance emerges on the other side. The absorbed light is usually transformed into other forms of energy, such as heat.
This phenomenon is crucial because it allows us to determine the concentration of substances in solutions using spectrophotometry. This process, simply put, involves shining light through a solution and measuring how much light is absorbed.
Substances absorb specific wavelengths of light, depending on their molecular structure. This leads to varying intensities of light when it exits the other side. By comparing the intensity of the initial light with the emerging light, we can derive important information about the solution, such as its concentration.

Concentration Calculation

The concentration of a sample in solution is a measure of how much solute is present per volume of solution, commonly expressed in moles per liter. In our spectrophotometry exercise, the concentration can be deduced using a specific mathematical formula:

• \[ C = -2500 \ln \left( \frac{I}{I_{0}} \right) \]

This equation quantifies concentration based on the natural logarithm of the ratio of the intensities of emergent and incident light.
To compute the concentration, you replace \(I\) and \(I_0\) with their respective values, simplify, and solve. This concentration calculation is vital for practical lab work, where determining the exact concentration is a foundational step in numerous experiments and analyses.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with base \(e\), where \(e\) is approximately equal to 2.718. The natural logarithm is useful when expressing time-dependent processes and is central to various scientific calculations, including those in chemistry and physics.
In the context of absorption and concentration computation, the use of the natural logarithm helps in linearizing exponential decay, which is a common pattern of absorption in spectrophotometry.
This inclusion simplifies complex multiplicative relationships into additive calculations, which are considerably easier to handle mathematically, making it possible to apply straightforward formulas like the one used to calculate concentration from light intensities. You perform these calculations using a scientific calculator or logarithm tables, ensuring the process is accessible even for students and laboratory technicians.

Intensity Measurement

Light intensity refers to the brightness or strength of light. In spectrophotometry, it's divided into two main components:

• Incident light intensity, \(I_0\), the light that first hits the sample.
• Emergent light intensity, \(I\), the light that has passed through the sample.

Measuring these values is essential for determining how much light has been absorbed by the sample, which in turn allows calculating the concentration.
The key tool used for these measurements is the spectrophotometer, an instrument designed to measure the amount of light that a sample absorbs at various wavelengths. This precision in measurement is critical to ensuring that the calculated concentrations are accurate and reliable for further studies or chemical analysis.
Accurate intensity measurement means more accurate absorption data, leading to precise concentration calculations. This accuracy is especially important in fields like pharmaceuticals, where small deviations in concentration can have significant effects.