Question

To obtain \(500 \mathrm{~kg}\) of glass composed of equimolar proportions of \(\mathrm{Na}_{2} \mathrm{SiO}_{3}\) and \(\mathrm{CaSiO}_{3}\), what weights of \(\mathrm{Na}_{2} \mathrm{CO}_{3}, \mathrm{CaCO}_{3}\), and \(\mathrm{SiO}_{2}\) should be used?

Short answer

To make 500 kg of glass composed of equimolar proportions of Na2SiO3 and CaSiO3, 111,366.78 g of Na2CO3, 105,063 g of CaCO3, and 126,075.6 g of SiO2 should be used.

Step-by-step solution

Calculate the amount in moles of Na2SiO3 and CaSiO3 to make 500 kg of glass

We know that the glass is composed of equimolar proportions of Na2SiO3 and CaSiO3, meaning equal number of moles. To find the amount in moles for each compound, we first determine the total molar mass of the glass. Molar Mass of Na2SiO3 = \(2 \times 23 + 28 + 3 \times 16\) = 122 g/mol Molar Mass of CaSiO3 = \(40 + 28 + 3 \times 16\) = 116 g/mol Total Molar Mass of Glass = Molar Mass of Na2SiO3 + Molar Mass of CaSiO3 Total Molar Mass of Glass = 122 g/mol + 116 g/mol = 238 g/mol Now, we'll find the amount in moles: Total moles of glass = \(\frac{500 \times 10^3}{238}\) = 2101.26 moles As the proportions are equimolar, half of the total moles will be Na2SiO3, and the other half will be CaSiO3: Moles of Na2SiO3 = Moles of CaSiO3 = \(\frac{2101.26}{2}\) = 1050.63 moles

Find the amount in moles of Na2CO3, CaCO3, and SiO2 required

From the balanced chemical equations, we can determine the amount in moles of Na2CO3, CaCO3, and SiO2 required. Na2CO3 + SiO2 -> Na2SiO3 + CO2 CaCO3 + SiO2 -> CaSiO3 + CO2 The stoichiometry suggests that for 1 mol of Na2SiO3, 1 mol of Na2CO3, and 1 mol of SiO2 are required. Similarly, for 1 mol of CaSiO3, 1 mol of CaCO3 and 1 mol of SiO2 are required. Moles of Na2CO3 = 1050.63 moles Moles of SiO2 = 1050.63 moles (for Na2SiO3) + 1050.63 moles (for CaSiO3) = 2101.26 moles Moles of CaCO3 = 1050.63 moles

Convert moles to weights of required chemicals

Now, we'll convert the moles of Na2CO3, CaCO3, and SiO2 to grams. Molar Mass of Na2CO3 = \(2 \times 23 + 12 + 3 \times 16\) = 106 g/mol Molar Mass of SiO2 = \(28 + 2 \times 16\) = 60 g/mol Molar Mass of CaCO3 = \(40 + 12 + 3 \times 16\) = 100 g/mol Weight of Na2CO3 = moles of Na2CO3 × Molar Mass of Na2CO3 Weight of Na2CO3 = \(1050.63 \times 106\) = 111,366.78 g Weight of SiO2 = moles of SiO2 × Molar Mass of SiO2 Weight of SiO2 = \(2101.26 \times 60\) = 126,075.6 g Weight of CaCO3 = moles of CaCO3 × Molar Mass of CaCO3 Weight of CaCO3 = \(1050.63 \times 100\) = 105,063 g Thus, to make 500 kg of glass composed of equimolar proportions of Na2SiO3 and CaSiO3, 111,366.78 g of Na2CO3, 105,063 g of CaCO3, and 126,075.6 g of SiO2 should be used.

Key concepts

Molar Mass Calculation

Understanding the concept of molar mass is crucial for solving stoichiometry problems in chemistry. Molar mass gives us the weight of one mole (approximately 6.022 x 10²³ particles) of a substance. This is usually expressed in grams per mole (g/mol).

To calculate the molar mass of a compound, add the molar masses of all the atoms in its chemical formula. For example, in the given exercise, the calculation for Na₂SiO₃ involved multiplying the atomic mass of sodium (Na) by 2 because there are two sodium atoms, adding the atomic mass of silicon (Si) once, and then adding the atomic mass of oxygen (O) multiplied by three since there are three oxygen atoms in the compound.

Key Steps for Molar Mass Calculation:

• Identify the number of each type of atom in the chemical formula.
• Look up the atomic mass for each element from the periodic table.
• Multiply the atomic mass of each element by the number of atoms of that element in the molecule.
• Add up all the individual masses to find the total molar mass.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry deals with the quantitative relationships between the reactants and products in a chemical reaction. It allows us to predict how much of each substance is needed or produced in a reaction.

To solve stoichiometry problems, it’s vital to work with a balanced chemical equation, which adheres to the law of conservation of mass. This law asserts that atoms are not created or destroyed in a chemical reaction.

Essential Principles of Stoichiometry:

• Use the coefficients in the balanced equation to understand the ratio of moles of reactants to moles of products.
• Convert mass of substances to moles or vice versa, depending on what the problem asks for, using the molar mass.
• Apply the mole ratios to find out the needed or produced amount of substances.

In the provided exercise, you can see how stoichiometry is used to calculate the moles of reactants required to produce a certain number of moles of the product.

Mole-to-Mass Conversion

Mole-to-mass conversion is a common task in chemistry that involves using a compound's molar mass to convert between the number of moles and the mass of a substance. You typically use the molar mass as a conversion factor to switch from moles to grams.

In practice, you multiply the number of moles of the substance by its molar mass to obtain the mass in grams. For example, in our textbook exercise, the calculated moles of Na₂CO₃ are converted to grams by multiplying by the molar mass of Na₂CO₃.

Steps for Mole-to-Mass Conversion:

• Determine the molar mass of the given substance.
• Use the formula: Mass (g) = Moles × Molar Mass (g/mol).
• Perform the multiplication to find the mass in grams.

This conversion is critical in preparing reactants for a chemical reaction to ensure exact stoichiometry and is fundamental in analytical chemistry for determining quantities of interest.