Problem 86 - Precalculus Mathematics for Calculus

Question

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 that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter) is found by 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 per liter.

Step 1: Understand the formula

The formula given is \(C=-2500 \ln \left(\frac{I}{I_{0}}\right)\), where \(C\) is the concentration of the sample. \(I_{0}\) is the initial light intensity, and \(I\) is the emergent light intensity.

Step 2: Substitute the known values

We know that \(I = 0.7 I_{0}\) because the intensity \(I\) is 70% of \(I_{0}\). Substitute \(\frac{I}{I_{0}} = 0.7\) into the formula for \(C\).

Step 3: Calculate the ratio

Substitute \(0.7\) into the concentration formula: \(C = -2500 \ln(0.7)\).

Step 4: Solve for the concentration

Use a calculator to find the natural logarithm of 0.7: \(\ln(0.7) \approx -0.3567\). Then, compute \(C = -2500 \, \times\, (-0.3567)\), which gives us \(C \approx 891.75\).

Key Concepts

Concentration Calculation

In spectrophotometry, determining the concentration of a substance in a solution involves a fascinating interplay of light and mathematics. The goal is to find out how much of a substance is present by observing how much light the substance absorbs. In our case, we utilize the formula:\[ C = -2500 \ln \left(\frac{I}{I_{0}}\right) \]Here, the variable \(C\) represents the concentration in moles per liter. By mathematical operations, we calculate \(C\) using the proportion of light that exits a sample compared to the initial amount of light introduced.Key Steps in Calculation:- Start by determining the ratio \(\frac{I}{I_{0}}\). It expresses the fraction of light remaining after passing through the sample.- In this case, knowing \(I\) is 70% of \(I_{0}\), we substitute \(0.7\) for \(\frac{I}{I_{0}}\).- Plug this ratio into the formula and solve for \(C\) using a calculator to handle the logarithmic function.After substituting and solving, the concentration calculation yields \(C \approx 891.75\). This value tells us the concentration of the sample in moles per liter.

Natural Logarithm

Natural logarithms are an integral part of calculating concentrations in spectrophotometry. They allow us to transform multiplicative relationships into additive ones, making complex calculations more accessible. The natural logarithm, denoted as \(\ln(x)\), has a base of \(e\), where \(e\) is an irrational number approximately equal to 2.718.

Understanding the Logarithmic Function

Computing the natural logarithm of a number, such as 0.7 in our example, essentially measures how much smaller the emergent light is compared to the incident light.- When we compute \(\ln(0.7)\), we find it to be approximately \(-0.3567\).- The negative value here indicates a loss of light intensity, typical since the medium absorbs some of the light.

Application in Concentration Calculation

In the formula used, the natural logarithm helps to linearize the concentration relationship. Without logarithms, this task would be significantly more complex and less intuitive. This approach makes it easier to derive clear and precise measurements of concentration from relatively simple measurements of light intensity.

Light Intensity

Light intensity is a fundamental aspect that defines how spectrophotometric analysis is possible. Intensity metrics allow us to deduce how much light is absorbed by the sample, thus inferring the concentration of substances.Light interacts with the sample by being directed into it with an initial intensity, \(I_0\). Some of the light is absorbed, and what's left exits the sample with intensity \(I\).

• Incident Light \(I_0\): The initial amount of light introduced to the sample.
• Emergent Light \(I\): The amount of light that exits the sample.

The fraction of light that persists is determined by comparing \(I\) with \(I_0\). In our exercise, it's provided that the emergent light is 70% of the incident light, or \(I = 0.7 I_0\).

Importance in Calculations

- Spectrophotometry hinges on these intensity values. They help compute the absorbed light quantity, which is inversely proportional to the substance concentration.- Without accurate intensity measurements, determining concentrations would be unreliable.Accurate light intensity readings assure us of the precision of our concentration results, vital in scientific research and various applications.