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 ?$

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

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

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 \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot 10\mathrm{H}_{2} \mathrm{O}\).