Problem 18 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A sample of sodium sulfate is known to be contaminated with copper ions. Analysis of reference solutions containing copper ions at known concentrations gave the absorbances shown in the table. Under the same conditions, a solution made by dissolving \(1.00 \mathrm{g}\) of the sodium sulfate sample in \(1.00 \mathrm{dm}^{3}\) distilled water gave an absorbance of \(0.20 .\) What was the purity of the sodium sulfate, assuming that copper ions are the only impurity? (Section 11.4 ) $$\begin{array}{llllll}\hline \text { Concentration of } \mathrm{Cu}^{2+} / \mathrm{ppm} & 2.0 & 4.0 & 6.0 & 8.0 & 10.0 \\\\\text {Absorbance } & 0.06 & 0.12 & 0.18 & 0.24 & 0.30 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The purity of the sodium sulfate is 99.3%.

Step 1: Understand the Problem

We need to determine the purity of the sodium sulfate sample by finding out how much copper ion is present in the solution. The given absorbance of 0.20 will help us determine the concentration of copper ions.

Step 2: Analyze the Data

We have two columns of data: concentration of copper ions ( [Cu$^{2+}$] in ppm) and their corresponding absorbance values. The relationship between concentration and absorbance is generally linear (following Beer's Law: A = εcl), allowing us to use this data to find the concentration at A = 0.20.

Step 3: Determine the Relationship

Observe that the absorbance changes linearly as concentration of copper ions changes. We can calculate the concentration of copper ions in the solution when absorbance is 0.20 by finding the proportional change between other data pairs.

Step 4: Apply Linear Relationship

To find the concentration of Cu$^{2+}$ at an absorbance of 0.20, we can calculate the slope (m) between points and use it for interpolation. For example, from 0.18 to 0.24 (6 ppm to 8 ppm), the slope is 3 ppm per 0.06 absorbance change.

Step 5: Calculate Copper Ion Concentration

Given the known slope of 3 ppm per 0.06 absorbance and knowing 0.18 corresponds to 6 ppm: \(\Delta A = 0.20 - 0.18 = 0.02\) and \(\Delta C = 3 \, \text{ppm} \times \frac{0.02}{0.06} = 1 \, \text{ppm}\). Thus, at 0.20 absorbance, concentration = 6 ppm + 1 ppm = 7 ppm.

Step 6: Determine Mass of Copper

To find the mass of Cu$^{2+}$ in 1 liter (since 1 g of the sodium sulfate sample is dissolved in 1 L), convert 7 ppm to mg: 7 ppm = 7 mg of Cu$^{2+}$ in 1 liter of solution.

Step 7: Calculate Purity

Subtract the mass of copper from the total mass of the sample to find the mass of sodium sulfate. Then, divide the mass of sodium sulfate by the original mass of the sample. Purity = \(\left(\frac{1.00 \, g - 0.007 \, g}{1.00 \, g}\right) \times 100\% = 99.3\%\).

Key Concepts

Beer's Law

In chemistry, Beer's Law is an essential principle used to understand how the absorbance of a light-absorbing material is related to its concentration in a solution. Beer's Law, often expressed as \( A = \varepsilon c l \), delineates that absorbance \( A \) is directly proportional to the concentration \( c \) of the absorbing species, the path length \( l \), and the molar absorptivity \( \varepsilon \). This equation underpins the technique of using spectrophotometry to assess how much of a substance is present in a solution, based on the light it absorbs. With Beer's Law, knowing the absorbance of a solution allows us to determine the concentration of an unknown sample, making it instrumental in purity analysis and chemical investigations.

Absorbance Measurement

Absorbance measurement is a fundamental aspect of spectrophotometry. In this process, a spectrophotometer measures how much light of a specific wavelength is absorbed as it passes through a solution. This measurement is expressed as "absorbance," denoted by the letter \( A \). The information gathered from this measurement is crucial for various analyses. In the context of this exercise, absorbance data help determine the concentration of copper ions in the solution. By recording the absorbance at different known concentrations, a relationship is established that helps us find unknown concentrations, validating the practical application of Beer's Law.

Copper Ion Concentration

The concentration of copper ions in a solution is a pivotal factor measured in parts per million (ppm) in our given scenario. By understanding the concentration, we can evaluate the degree of contamination in a sample. In the solution of interest, the absorbance due to these copper ions is measured at 0.20. By referring to a reference table linking known concentrations of copper ions to absorbance, an estimation of the copper ion concentration in this unknown sample can be made. When the absorbance reading falls between two known values, techniques like interpolation become invaluable in interpolating the precise concentration.

Linear Interpolation

Linear interpolation is a mathematical method employed to estimate unknown values that fall within the range of two known values. In this exercise, linear interpolation is utilized to determine the concentration of copper ions corresponding to an absorbance of 0.20. Given the linearly increasing nature of the absorbance as presented in reference data, interpolation enables us to deduce the unknown concentration efficiently. By calculating the slope and considering the proportional change, the concentration at the measured absorbance can be precisely estimated, bridging the gap between experimental data and theoretical calculations.

Contaminant Analysis

In the realm of chemistry, contaminant analysis involves identifying and quantifying undesired substances in a sample. In our exercise, the aim is to assess copper ion contamination within a sodium sulfate sample. Through the methodical use of Beer's Law and absorbance measurements, the concentration of copper ions—a potential contaminant—reveals insight into the sample's purity. Accurate contaminant analysis not only tells us about the proportion of the impurity present but also informs purity calculations, crucial for applications where chemical integrity is paramount. Understanding these concentrations allows for determining the efficacy and safety of chemical samples.