Problem 59 - Chemistry The Central Science In Si Units

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 \(2.558-g\) sample of washing soda is heated at \(125^{\circ} \mathrm{C},\) all the water of hydration is lost, leaving \(0.948 \mathrm{~g}\) of \(\mathrm{Na}_{2} \mathrm{CO}_{3} .\) What is the value of \(x ?\)

Step-by-Step Solution

Answer

The value of \(x\) is 10.

Step 1: Determine Mass of Water Lost

First, we find the mass of water lost when washing soda is heated by subtracting the mass of anhydrous sodium carbonate from the mass of the hydrated compound. The mass of hydrated \(\mathrm{Na}_2\mathrm{CO}_3 \cdot x \mathrm{H}_2\mathrm{O}\) is \(2.558 \, \text{g}\), and the mass of \(\mathrm{Na}_2\mathrm{CO}_3\) is \(0.948 \, \text{g}\). So, the mass of water lost is \(2.558 \, \text{g} - 0.948 \, \text{g} = 1.610 \, \text{g}.\)

Step 2: Calculate Moles of Anhydrous Sodium Carbonate

Calculate the number of moles of sodium carbonate. The molar mass of \(\mathrm{Na}_2\mathrm{CO}_3\) is \(106 \, \text{g/mol}\). Using the formula, \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\), we find \[\text{moles of } \mathrm{Na}_2\mathrm{CO}_3 = \frac{0.948 \, \text{g}}{106 \, \text{g/mol}} = 0.00894 \, \text{mol}.\]

Step 3: Calculate Moles of Water

Now calculate the moles of water lost. The molar mass of water, \(\mathrm{H}_2\mathrm{O}\), is \(18 \, \text{g/mol}\). Using the same formula, \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\), we find \[\text{moles of } \mathrm{H}_2\mathrm{O} = \frac{1.610 \, \text{g}}{18 \, \text{g/mol}} = 0.0894 \, \text{mol}.\]

Step 4: Determine the Ratio of Moles

To find \(x\), the number of moles of water per mole of sodium carbonate, divide the moles of water by the moles of \(\mathrm{Na}_2\mathrm{CO}_3\). The calculation is \[x = \frac{0.0894 \, \text{mol of } \mathrm{H}_2\mathrm{O}}{0.00894 \, \text{mol of } \mathrm{Na}_2\mathrm{CO}_3} = 10.\] Thus, \(x = 10\).

Key Concepts

Moles Calculation

Moles are a fundamental concept in chemistry that allow us to count particles at the atomic level. When we perform calculations involving chemical compounds, we often need to know how many moles of each substance we are dealing with.
To calculate moles, we use the formula:

• \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \)

The mass is usually given in grams, while the molar mass of a compound is the mass of one mole of the compound, often found using the periodic table.
This formula is vital for converting between grams and moles, which is essential for balancing chemical equations and determining the relationships between substances in chemical reactions.

Sodium Carbonate

Sodium carbonate, often referred to as washing soda, is a white, water-soluble salt used in various household, industrial, and chemical processes. Its chemical formula is \( \mathrm{Na}_2\mathrm{CO}_3 \), indicating that each molecule has two sodium (Na) atoms, one carbon (C) atom, and three oxygen (O) atoms.
Sodium carbonate is commonly encountered in its hydrated form, which means the compound includes a specific number of water molecules bound to it. These water molecules are not just randomly associated; they are part of the crystalline structure of the compound.
The anhydrous form (no water molecules) has a molar mass of 106 g/mol, which is used in calculations to determine quantities in reactions or when converting to moles.

Water of Hydration

Water of hydration refers to water molecules that are chemically bound within a crystalline structure of a compound. This means that the compound includes these water molecules as part of its formula.
For example, washing soda is known as sodium carbonate hydrate and is often represented as \( \mathrm{Na}_2\mathrm{CO}_3 \cdot x \mathrm{H}_2\mathrm{O} \). The \( x \) represents the number of water molecules bound to a single molecule of sodium carbonate.
When such hydrates are heated, the water is driven off, leaving behind the anhydrous form of the compound. Understanding the water of hydration is crucial in chemistry because it affects the properties and reactivity of the compound.

Molar Mass

Molar mass is the mass of one mole of a substance, typically given in grams per mole (g/mol). It is an essential concept in chemistry because it allows us to convert between the mass of a substance and the number of moles.
To find the molar mass, one must add the atomic masses of all atoms present in the compound's formula. For sodium carbonate ( \( \mathrm{Na}_2\mathrm{CO}_3 \), you would calculate it as follows:

• 2 sodium atoms: \( 2 \times 23 \text{g/mol} = 46 \text{g/mol} \)
• 1 carbon atom: \( 1 \times 12 \text{g/mol} = 12 \text{g/mol} \)
• 3 oxygen atoms: \( 3 \times 16 \text{g/mol} = 48 \text{g/mol} \)

Adding these up gives the molar mass of \( \mathrm{Na}_2\mathrm{CO}_3 \) as 106 g/mol.
This value is critical when performing mole calculations, as it directly translates mass measurements into moles, which can then be used to explore other chemical properties and reactions.