Problem 132 - Chemistry The Molecular Science

Question

A mixture contains only \(\mathrm{MgSO}_{4}\) and \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} .\) If the mass percent of \(\mathrm{MgSO}_{4}\) in the mixture is \(32.0 \%,\) what is the mass percent of sulfate in the mixture?

Step-by-Step Solution

Answer

The sulfate mass percent in the mixture is approximately 75.0%.

Step 1: Understand the Compounds

There are two compounds in the mixture: \(\mathrm{MgSO}_{4}\) and \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} \). Both contain sulfate ions (\(\mathrm{SO}_4^{2-} \)). We need to find out the sulfate percentage in the mixture.

Step 2: Calculate Molar Mass of Compounds

Calculate the molar mass of \(\mathrm{MgSO}_{4}\) which is \(24.31 + 32.07 + 4\times16.00 = 120.37\) g/mol. For \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\), the molar mass is \(2\times14.01 + 8\times1.01 + 32.07 + 4\times16.00 = 132.14\) g/mol.

Step 3: Calculate Sulfate's Contribution to Each Compound

The molar mass of sulfate \(\mathrm{SO}_4\) is \(32.07 + 4\times16.00 = 96.07\) g/mol. Therefore, in \(\mathrm{MgSO}_{4}\), sulfate's contribution is \(\frac{96.07}{120.37} \approx 0.798\) or 79.8%. In \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\), sulfate's contribution is \(\frac{96.07}{132.14} \approx 0.727\) or 72.7%.

Step 4: Determine Sulfate Percentage in Mixture

Let \(x\) be the mass of \(\mathrm{MgSO}_{4}\) and \(x+\frac{68}{32}x = \frac{100}{32}x\) be the total mass of the mixture. The sulfate mass percent in the mixture is \[ 0.32 \times 79.8\% + 0.68 \times 72.7\% = 25.536\% + 49.436\% = 74.972\% \].

Step 5: Final Sulfate Mass Percent

From the above calculation, the mass percent of sulfate in the mixture is approximately 75.0%.

Key Concepts

Understanding Sulfate Ions

When dealing with chemical mixtures, it is important to grasp the role of specific ions such as sulfate ions, represented by the formula \(\text{SO}_4^{2-}\). These ions are made up of one sulfur atom and four oxygen atoms. The presence of sulfate ions can significantly affect the properties and reactions of the compounds they are part of.

Sulfate ions are negatively charged, which makes them anions. In many chemical compounds, they commonly bond with positive ions, such as magnesium ions in \(\text{MgSO}_4\) or ammonium ions in \(\left(\text{NH}_4\right)_2\text{SO}_4\). This bonding helps to maintain electrical neutrality in the compound.

Understanding the role and characteristics of sulfate ions is crucial for determining their contribution to the molar mass and percentage composition in chemical mixtures.

Calculating Molar Mass

To solve problems involving chemical compounds, it is vital to calculate the molar mass accurately. The molar mass is the sum of the atomic masses of all atoms in a compound, measured in grams per mole \(\text{g/mol}\).

For \(\text{MgSO}_4\), the molar mass calculation involves adding the atomic masses of magnesium \(\text{Mg}\), sulfur \(\text{S}\), and oxygen \(\text{O}\). Specifically, \(24.31\ \text{(Mg)} + 32.07\ \text{(S)} + 4 \times 16.00\ \text{(O)}\), resulting in a molar mass of \(120.37\ \text{g/mol}\).

Similarly, for \(\left(\text{NH}_4\right)_2\text{SO}_4\), add the masses of nitrogen \(\text{N}\), hydrogen \(\text{H}\), sulfur, and oxygen. This gives \(2\times14.01\ \text{(N)} + 8\times1.01\ \text{(H)} + 32.07\ \text{(S)} + 4\times16.00\ \text{(O)}\), totaling \(132.14\ \text{g/mol}\). Calculating the molar mass helps in understanding the contribution of individual ions, like sulfate, to the compound’s total mass.

Overview of Chemical Compounds

Chemical compounds are substances formed by the combination of two or more different types of atoms. These compounds have distinct chemical properties that are different from their constituent elements.

For example, \(\text{MgSO}_4\), also known as magnesium sulfate, is a compound that combines magnesium, sulfur, and oxygen atoms. Each element contributes specific properties to the compound, such as solubility or reactivity.

Similarly, \(\left(\text{NH}_4\right)_2\text{SO}_4\), known as ammonium sulfate, combines ammonium and sulfate ions. Understanding these compounds requires recognizing their chemical formula, which represents the ratio of atoms in the compound.

In chemical analysis, it is crucial to determine the mass percentages of different ions within these compounds. This involves understanding the composition and interactions of the elements within the compounds. Such information is paramount when assessing substances like mixtures containing \(\text{MgSO}_4\) and \(\left(\text{NH}_4\right)_2\text{SO}_4\).