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 \(x ?\)

Short answer

The value of x in the formula \(\mathrm{Na}_2\mathrm{CO}_3 \cdot x\mathrm{H}_2\mathrm{O}\) for washing soda hydrate is 10, making the formula \(\mathrm{Na}_2\mathrm{CO}_3\cdot10\mathrm{H}_2\mathrm{O}\).

Step-by-step solution

Calculate the moles of \(\mathrm{Na}_2\mathrm{CO}_3\) after heating

First, we need to find the moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) using the given mass after heating (0.948 g) and the molar mass of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\). The molar mass of \(\mathrm{Na}_2\mathrm{CO}_3\) is \(2\times23.0 (\mathrm{Na}) + 12.0 (\mathrm{C}) + 3\times16.0 (\mathrm{O}) = 106.0 \,\mathrm{g/mol}\). Now, we can calculate the moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) as: \[\frac{0.948\;\text{g}}{106.0\;\text{g/mol}} = 0.00894\;\text{mol}\]

Calculate the mass of the water of hydration released

To find the mass of the water of hydration, we subtract the mass of \(\mathrm{Na}_2\mathrm{CO}_3\) after heating from the mass of the hydrate before heating: \[2.558 \; \text{g} - 0.948 \;\text{g} = 1.610 \;\text{g}\]

Calculate the moles of water of hydration released

Now, we need to find the moles of the water of hydration released. To do this, we will use the mass of water released (1.610 g) and the molar mass of water, which is \(2\times1.0 (\mathrm{H}) + 16.0 (\mathrm{O}) = 18.0 \,\mathrm{g/mol}\). Now, we can calculate the moles of \(\mathrm{H}_2\mathrm{O}\) as: \[\frac{1.610\;\text{g}}{18.0\;\text{g/mol}} = 0.0894\;\text{mol}\]

Determine the value of \(x\) by calculating the ratio of moles

We now have the moles of \(\mathrm{Na}_2\mathrm{CO}_3\) (0.00894 mol) and the moles of \(\mathrm{H}_2\mathrm{O}\) (0.0894 mol) present in the washing soda. To find the value of x, we need to find the ratio of the moles of \(\mathrm{H}_2\mathrm{O}\) to moles of \(\mathrm{Na}_2\mathrm{CO}_3\): \[x = \frac{\text{moles of }\mathrm{H}_2\mathrm{O}}{\text{moles of }\mathrm{Na}_2\mathrm{CO}_3} = \frac{0.0894\;\text{mol}}{0.00894\;\text{mol}} = 10\] So, the value of x is 10, and the formula of the washing soda hydrate is \(\mathrm{Na}_2\mathrm{CO}_3\cdot10\mathrm{H}_2\mathrm{O}\).

Key concepts

Molar Mass Calculation

When dealing with compounds like washing soda, calculating molar mass is essential. The molar mass of a substance is the mass of one mole of its molecules. This is crucial in determining how many moles are present in a given mass.
For example, the molar mass of sodium carbonate (\(\mathrm{Na}_2\mathrm{CO}_3\)) is calculated by adding up the atomic masses of all atoms in its formula. Sodium (Na) has an atomic mass of approximately 23.0 g/mol, carbon (C) is around 12.0 g/mol, and oxygen (O) is about 16.0 g/mol.

• There are 2 sodium atoms: \(2 \times 23.0 = 46.0\) g/mol.
• One carbon atom: \(1 \times 12.0 = 12.0\) g/mol.
• Three oxygen atoms: \(3 \times 16.0 = 48.0\) g/mol.

Adding these together, the molar mass of \(\mathrm{Na}_2\mathrm{CO}_3\) is \(106.0\,\mathrm{g/mol}\). Knowing such calculations helps in further determinations in chemistry, especially with hydrates.

Hydrate Chemistry

In chemistry, hydrates are compounds that incorporate water molecules into their solid structure. These water molecules are known as 'water of hydration.' They are not simply mixed with the compound but are chemically bonded, altering the compound's physical properties.
The formula \(\mathrm{Na}_2\mathrm{CO}_3 \cdot x \mathrm{H}_2\mathrm{O}\) indicates that sodium carbonate is a hydrate with \(x\) moles of water per mole of sodium carbonate. When hydrates are heated, they often release water vapor, leaving behind the anhydrous form.
Understanding hydrates involves:

• Recognizing their formulas, such as \(\mathrm{Na}_2\mathrm{CO}_3 \cdot 10 \mathrm{H}_2\mathrm{O}\)
• Identifying what happens during the heating process, where water is lost.

By studying hydrates, one can learn about the interactions and importance of water in chemical structures.

Molecule Ratio Determination

In a hydrate, determining the ratio of water molecules to the rest of the compound is crucial for understanding its chemical structure. This requires calculating the number of moles of water and the compound and comparing them.
From the original problem, the number of moles of \(\mathrm{Na}_2\mathrm{CO}_3\) was calculated earlier from the mass after heating. The moles of water were found from the mass of water that was lost.
To find the value of \(x\), the ratio of the moles of water to the moles of sodium carbonate is calculated:

• Determine moles of water: measured by mass / molar mass, e.g., \(1.610\,\text{g} / 18.0\,\text{g/mol} = 0.0894\,\text{mol}\)
• Determine moles of \(\mathrm{Na}_2\mathrm{CO}_3\): e.g., \(0.948\,\text{g} / 106.0\,\text{g/mol} = 0.00894\,\text{mol}\)
• Calculate the ratio \(x = \frac{0.0894\,\text{mol}}{0.00894\,\text{mol}} = 10\)

This ratio reveals that there are 10 water molecules per molecule of \(\mathrm{Na}_2\mathrm{CO}_3\), indicating the hydrate's full formula as \(\mathrm{Na}_2\mathrm{CO}_3 \cdot 10 \mathrm{H}_2\mathrm{O}\).