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 \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot \mathrm{xH}_{2} \mathrm{O},\) 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 \(\chi ?\)

Short answer

The value of x in the washing soda formula \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot \mathrm{xH}_{2} \mathrm{O}\) is approximately 10. Therefore, the washing soda formula can be written as \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot 10\mathrm{H}_{2} \mathrm{O}\).

Step-by-step solution

Find the mass of water in the original sample

The mass of water in the original sample can be determined by subtracting the mass of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) from the initial mass of the washing soda. Total mass of sample = 2.558 g Mass of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) = 0.948 g Mass of water = 2.558 g - 0.948 g Mass of water = 1.610 g

Calculate the moles of anhydrous \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) and H\(_{2}\)O

Now we need to calculate the moles of anhydrous \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) and H\(_{2}\)O in the sample. The molar mass of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) = (2 x 22.99 g/mol) + (1 x 12.01 g/mol) + (3 x 16.00 g/mol) = 105.99 g/mol Moles of anhydrous \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) = mass of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) / molar mass of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) Moles of anhydrous \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) = 0.948 g / 105.99 g/mol = 0.008950 mol The molar mass of water (H\(_{2}\)O) = (2 x 1.01 g/mol) + (1 x 16.00 g/mol) = 18.02 g/mol Moles of water (H\(_{2}\)O) = mass of water / molar mass of water Moles of water (H\(_{2}\)O) = 1.610 g / 18.02 g/mol = 0.08939 mol

Calculate the value of x

Now, we need to calculate the value of x in the washing soda formula \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot \mathrm{xH}_{2} \mathrm{O},\) by using the stoichiometric ratio. x = moles of water (H\(_{2}\)O) / moles of anhydrous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) x = 0.08939 mol / 0.008950 mol x ≈ 10 Now we know the value of x in the washing soda formula, which is approximately 10. Therefore, the washing soda formula can be written as \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot 10\mathrm{H}_{2} \mathrm{O}\).

Key concepts

Chemical Formulas

Chemical formulas provide a simple way to represent the elements contained in a compound and their respective quantities. They are fundamental in understanding the composition and structure of substances we deal with in chemistry. These formulas indicate the types and numbers of atoms bonded together. For example, the formula for water, \(\mathrm{H}_{2} \mathrm{O}\), tells us that each molecule of water consists of two hydrogen atoms and one oxygen atom. When it comes to ionic compounds like washing soda \(\mathrm{Na}_2 \mathrm{CO}_3\), chemical formulas provide insights into the composition of ions. Here, sodium carbonate includes two sodium ions and one carbonate ion. Chemical formulas also extend to hydrates, where water molecules are integral to the solid's structure. The number following the 'dot' in formulas like \(\mathrm{Na}_2 \mathrm{CO}_3 \cdot \mathrm{xH}_2 \mathrm{O}\) denotes how many water molecules associate with the formula unit in the crystal lattice. By understanding chemical formulas, you can grasp how elements interrelate in compounds, predict the properties of the compound, and even figure out significant reaction pathways.

Hydrates

Hydrates are compounds that incorporate water molecules into their crystalline structure. These water molecules, known as water of crystallization, play a crucial role in defining the physical properties of the compound. When represented in a chemical formula, a dot separates the main component of a compound from the associated water, as seen in \(\mathrm{Na}_2 \mathrm{CO}_3 \cdot 10\mathrm{H}_2 \mathrm{O}\). The number of water molecules attached to each formula unit in a hydrate is specified by the subscript next to the water molecule, here represented by '10'. This specifies that for every one unit of sodium carbonate, there are 10 water molecules integrated into the crystal structure. To determine the number of water molecules in a hydrate, you must often heat the compound to remove the water. This process converts the hydrate to an anhydrous form, as demonstrated when washing soda is heated to expel its water content. The difference in mass before and after heating allows the calculation of water's contribution, a vital step in determining the hydrate formulas.

Mole Concept

The mole concept is a central pillar of chemistry, essential for quantifying substances in chemical reactions. The mole is akin to a counting unit, much like a dozen, but much larger, defined as containing exactly \(6.022 \times 10^{23}\) particles, be they atoms, molecules, or ions. This huge number, known as Avogadro's number, helps bridge the tiny world of atoms and the quantities of material we use in the lab. In practical applications, the mole allows chemists to calculate how much of a material to use or expect from a reaction. For instance, to find moles from a mass, you divide by the substance's molar mass, which aggregates the atomic masses of each element in the compound. In the example of washing soda,

• the molar mass of \(\mathrm{Na}_2 \mathrm{CO}_3\) is calculated as \(105.99\, \text{g/mol}\)
• while for water \(\mathrm{H}_2 \mathrm{O}\), it’s \(18.02\, \text{g/mol}\).

This enables determining the moles of each component, thereby solving problems like finding 'x' in the formula \(\mathrm{Na}_2 \mathrm{CO}_3 \cdot \mathrm{xH}_2 \mathrm{O}\). Understanding the mole concept ensures precise measurements and provides insights into the stoichiometric relationships within reactions, enhancing the predictability and effectiveness of chemical processes.