Problem 107 - General Chemistry: Principles and Modern Applications

Question

A thoroughly dried 1.271 g sample of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is exposed to the atmosphere and found to gain \(0.387 \mathrm{g}\) in mass. What is the percent, by mass, of \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\) in the resulting mixture of anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and the decahydrate?

Step-by-Step Solution

Answer

The percent, by mass, of \( \mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O} \) in the resulting mixture is approximately 76.68%.

Step 1: Calculate Mass of Hydrate

First calculate the mass of the resulting hydrate. To do this, add the mass of the anhydrous compound (\(1.271 \, g\)) to the gain in mass caused by the absorption of water (\(0.387 \, g\)).

Step 2: Calculate Molecular Weight

Calculate the molar mass of anhydrous \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and of \(\mathrm{H}_{2} \mathrm{O}\). The molar mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is approximately 142.04 g/mol and that of \(\mathrm{H}_{2} \mathrm{O}\) is approximately 18.015 g/mol. Therefore, the molar mass of the hydrate \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\) is \(142.04 \, g/mol + 10 \times 18.015 \, g/mol = 322.19 \, g/mol\).

Step 3: Calculate Mass Percentage

To find the mass percentage of the hydrate, divide the mass of the anhydrous compound by the mass of the hydrate and multiply by 100. The mass percentage of the hydrate is then \(\frac{1.271 \, g}{1.658 \, g} \times 100 = 76.68\%\).

Key Concepts

Understanding Hydration in Chemistry

In chemistry, the term "hydration" refers to the process by which a substance absorbs water molecules and integrates them into its structure. This often forms a hydrous compound known as a hydrate. Hydrates are prevalent within many chemical processes and are essential to understand as they often impact a compound's properties and behavior.

When a compound like anhydrous sodium sulfate (\(\mathrm{Na}_{2} \mathrm{SO}_{4}\)) is exposed to moisture, water molecules can attach to its structure, forming the hydrated form of the compound—in this specific case, sodium sulfate decahydrate (\(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10a \mathrm{H}_{2} \mathrm{O}\)).

Hydrates often have defined ratios of water to the compound, which can be derived or determined through experimentation, such as the gain in mass seen when an anhydrous sample absorbs water. Understanding this concept is crucial in various practical and theoretical applications of chemistry, from basic laboratory settings to industrial processes.

Molar Mass Calculation: Getting the Weights Right

Accurate molar mass calculation is foundational for problems involving chemical formulas and reactions. Each element in a compound contributes a specific atomic mass, and understanding how to sum these accurately is key.

For sodium sulfate (\(\mathrm{Na}_{2} \mathrm{SO}_{4}\)), the molar mass is calculated by adding the atomic masses of all constituent atoms:

• 2 sodium (Na) atoms: 2 x 22.99 g/mol = 45.98 g/mol
• 1 sulfur (S) atom: 32.07 g/mol
• 4 oxygen (O) atoms: 4 x 16.00 g/mol = 64.00 g/mol

Adding these together gives \(142.04 \text{ g/mol}\) for \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) alone. The water of hydration also matters and must be included for complete compounds like \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\). Water has a molar mass of \(18.015grams/mol\). Thereby, a hydrate with 10 water molecules has a molar mass addition of \(180.15 \text{ grams/mol}\), making the total \(322.19 \text{ g/mol}\) for the hydrated compound.

These calculations allow chemists to understand how much of each component exists in a chemical formula or reaction, aiding in precise experimental and industrial formulations.

Mass Percentage - A Measure of Composition

The concept of mass percentage is crucial when analyzing chemical mixtures as it provides a straightforward way to express concentration or composition of components. Mass percentage (\(\%\)) indicates how much of a substance’s mass is contributed by a specific component within a larger mixture.

This calculation is executed using the formula:\[\text{Mass percentage} = \left( \frac{\text{Mass of Component}}{\text{Total Mass of Mixture}} \right) \times 100\]In the case of sodium sulfate hydrate, you might determine mass percentage by finding how much of the total weight is due to the anhydrous sodium sulfate versus the hydrated form.

Once the samples are exposed to moisture, calculating the percentage of \(\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}\) becomes straightforward:

• Total mass of the hydrated sample: initial mass + moisture gained (\(1.271 \text{ g} + 0.387 \text{ g} = 1.658 \text{ g}\)).
• The mass percentage of the hydrate is then calculated as \(\frac{1.271 \text{ g}}{1.658 \text{ g}} \times 100 \approx 76.68\%\).

Understanding mass percentage helps ascertain component proportion in chemical mixtures, which is vital for diverse fields like pharmaceuticals, materials science, and many chemical industries.