Question

Washing soda, a compound used to prepare hard water for washing laundry, is a hydrate, which means that a certain number of water molecules are included in the solid structure. Its formula can be written as \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot x \mathrm{H}_{2} \mathrm{O}\), where \(x\) is the number of moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). When a \(2.558-\mathrm{g}\) sample of washing soda is heated at \(25^{\circ} \mathrm{C}\), all the water of hydration is lost, leaving \(0.948 \mathrm{~g}\) of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). What is the value of \(x\) ?

Short answer

The value of \(x\) is approximately 10, which means there are 10 moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). So the formula for washing soda is \(\mathrm{Na_{2}CO_3·10 H_{2}O}\).

Step-by-step solution

Convert masses to moles

To convert the mass of a substance to moles, we use the formula: Moles = \(\frac{\text{mass}}{\text{molar mass}}\) Molar Mass of \(\mathrm{Na_{2}CO_3}\) = \((2 \times 22.99) + (12.01) + (3 \times 16.00)\) g/mol = \(105.98\) g/mol Mass of anhydrous \(\mathrm{Na_{2}CO_3}\) = \(0.948\) g Moles of anhydrous \(\mathrm{Na_{2}CO_3}\) = \(\frac{0.948 \ \text{g}}{105.98 \ \text{g/mol}}\) = \(0.00894\) mol

Calculate moles of water lost

Mass of washing soda before heating = \(2.558\) g Mass of washing soda after heating = \(0.948\) g Mass of water lost = Mass before heating - Mass after heating = \(2.558 \ \text{g} - 0.948 \ \text{g}\) = \(1.610\) g Molar Mass of \(\mathrm{H_{2}O}\) = \((2 \times 1.01) + (1 \times 16.00)\) g/mol = \(18.02\) g/mol Moles of water lost = \(\frac{1.610 \ \text{g}}{18.02 \ \text{g/mol}}\) = \(0.0893\) mol

Determine the value of x

Now, we will divide the moles of water lost by the moles of anhydrous \(\mathrm{Na_{2}CO_3}\): x = \(\frac{\text{moles of water lost}}{\text{moles of anhydrous Na}_{2}\text{CO}_{3}} = \frac{0.0893 \ \text{mol}}{0.00894 \ \text{mol}} = 9.99\) Since \(x\) represents the number of moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\), we can round \(x\) to the nearest integer. x ≈ 10 Therefore, the value of \(x\) is 10, which means there are 10 moles of \(\mathrm{H}_{2} \mathrm{O}\) per mole of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). The formula for washing soda is \(\mathrm{Na_{2}CO_3·10 H_{2}O}\).

Key concepts

Hydrate compounds

Hydrate compounds are fascinating chemical structures that include water molecules as part of their crystalline form. When these compounds are heated, the water is often released, transforming the hydrate into an anhydrous form. This process is essential in determining the composition of many compounds like washing soda, commonly used to treat hard water.

In hydrate compounds, the water molecules are integrated into the solid structure in specific ratios that depend on the substance. These water molecules are crucial not only for maintaining the structure but also for the physical properties of the compound. For example, in washing soda (\(\mathrm{Na_{2}CO_3\cdot xH_{2}O}\)), the water content determines its ability to dissolve minerals in hard water, acting as a water softener.

Understanding how to manipulate these water molecules allows chemists to create solutions and remedies for everyday problems like hard water. When studying or working with hydrate compounds, it's important to consider the number of water molecules involved, often represented by the variable \(x\) in the chemical formula. This highlights the pivotal role that water plays in determining the compound's characteristics.

Molar mass calculations

Molar mass calculations are key in stoichiometry, allowing chemists to convert between the mass of a substance and the number of moles. This conversion is essential for determining how many moles are present in a given mass, which is crucial in quantifying chemical reactions.

Calculating the molar mass involves summing up the atomic weights of each element in the compound. For washing soda (\(\mathrm{Na_2CO_3}\)), this involves adding the atomic masses of sodium (\(2 \times 22.99\) g/mol), carbon (\(12.01\) g/mol), and oxygen (\(3 \times 16.00\) g/mol), which results in a molar mass of \(105.98\) g/mol.

Once the molar mass is known, you can determine the moles from a given mass using the formula: \[ \text{Moles} = \frac{\text{mass}}{\text{molar mass}} \] This calculation tells us how much of a substance we have in terms of atomic quantity rather than just weight, making it easier to understand and predict the outcomes of chemical reactions.

Chemical formulas

Chemical formulas provide a shorthand representation of the composition of molecules and compounds. They indicate the elements present and the number of each type of atom. These formulas are crucial for understanding the ratios of atoms in compounds.

For example, the formula for washing soda can be noted as \(\mathrm{Na_2CO_3\cdot10H_2O}\), where ten is the number of water molecules per formula unit. This concise notation helps convey complex chemical structures in a straightforward manner, allowing chemists to communicate and record chemical reactions and properties efficiently.

In stoichiometry problems, chemical formulas help identify the reactants and products as well as their proportions in a reaction. Knowing how to interpret and use these formulas is essential for performing calculations related to reactions, such as determining how much product is formed from given reactants or, inversely, how much reactant is needed to produce a desired amount of product.

• Represents exact atom count in a compound.
• Indicates the ratios in chemical reactions.
• Essential for calculating moles and mass in reactions.

Understanding chemical formulas and the information they convey is foundational for studying and applying chemistry in practical and theoretical contexts.