Problem 63 - Chemistry The Molecular Science

Question

Calculate the volume of concentrated \((98 \%)\) sulfuric acid that is needed to produce two tons of phosphoric acid from the reaction of sulfuric acid with sufficient phosphate-bearing rock. Density of conc. sulfuric acid = \(1.84 \mathrm{~g} / \mathrm{mL}\) \(\mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2}(\mathrm{~s})+3 \mathrm{H}_{2} \mathrm{SO}_{4}(\ell) \longrightarrow 2 \mathrm{H}_{3} \mathrm{PO}_{4}(\ell)+3 \mathrm{CaSO}_{4}(\mathrm{~s})\)

Step-by-Step Solution

Answer

You need approximately 1,663.71 liters of concentrated sulfuric acid.

Step 1: Write the balanced chemical equation

The given balanced chemical equation is: \[ \text{Ca}_3(\text{PO}_4)_2 + 3\text{H}_2\text{SO}_4 \rightarrow 2\text{H}_3\text{PO}_4 + 3\text{CaSO}_4 \] This equation shows that 3 moles of sulfuric acid produce 2 moles of phosphoric acid.

Step 2: Convert phosphoric acid mass to moles

Two tons of phosphoric acid is equivalent to 2000 kg. The molar mass of phosphoric acid \((\text{H}_3\text{PO}_4)\) is 98 g/mol.First, convert the mass to grams: \[ \text{2000 kg} \times 1000 = 2,000,000 \text{ grams} \]Now, calculate the moles:\[ \text{Moles of } \text{H}_3\text{PO}_4 = \frac{2,000,000 \text{ grams}}{98 \text{ g/mol}} = 20,408.16 \text{ moles} \]

Step 3: Calculate moles of sulfuric acid

From the balanced equation, 3 moles of \(\text{H}_2\text{SO}_4\) produce 2 moles of \(\text{H}_3\text{PO}_4\).So:\[ \text{Moles of } \text{H}_2\text{SO}_4 = \frac{20,408.16 \times 3}{2} = 30,612.24 \text{ moles} \]

Step 4: Find mass of sulfuric acid needed

The molar mass of sulfuric acid \((\text{H}_2\text{SO}_4)\) is 98 g/mol, same as phosphoric acid.Calculate the mass:\[ \text{Mass of } \text{H}_2\text{SO}_4 = 30,612.24 \times 98 = 2,999,999.52 \text{ grams} \]Converting to kilograms:\[ 2,999,999.52 \text{ grams} = 3000 \text{ kg} \]

Step 5: Adjust for sulfuric acid concentration

Since the sulfuric acid is 98% concentrated, adjust the mass calculation:\[ \text{Actual mass } = \frac{3000}{0.98} = 3061.22 \text{ kg} \]

Step 6: Convert mass to volume

The density of concentrated sulfuric acid is 1.84 g/mL. Convert the actual mass to grams:\[ 3061.22 \text{ kg} = 3,061,220 \text{ g} \]Then, calculate volume:\[ \text{Volume} = \frac{3,061,220}{1.84} = 1,663,706.52 \text{ mL} \]Convert to liters:\[ 1,663,706.52 \text{ mL} = 1,663.71 \text{ L} \]

Step 7: Summary

To produce two tons of phosphoric acid, you need approximately 1,663.71 liters of concentrated sulfuric acid.

Key Concepts

Balanced Chemical Equation

Understanding a balanced chemical equation is crucial in the field of chemical stoichiometry. It allows us to understand the ratio in which various chemicals react or are produced. The equation is said to be balanced when the number of atoms for each element is equal on both sides of the reaction. This conserves the mass as dictated by the law of conservation of mass.
In our particular example, the equation is: \[\text{Ca}_3(\text{PO}_4)_2 + 3\text{H}_2\text{SO}_4 \rightarrow 2\text{H}_3\text{PO}_4 + 3\text{CaSO}_4\]This equation tells us that three moles of sulfuric acid (\(\text{H}_2\text{SO}_4\)) react with one mole of phosphate-bearing rock (\(\text{Ca}_3(\text{PO}_4)_2\)) to produce two moles of phosphoric acid (\(\text{H}_3\text{PO}_4\)) and three moles of calcium sulfate (\(\text{CaSO}_4\)). This molecular "recipe" is what you need to follow when calculating quantities in reactions.

Molar Mass

Molar mass is the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and allows us to convert between mass and moles.
For instance, phosphoric acid (\(\text{H}_3\text{PO}_4\)) has a molar mass of 98 g/mol. This means that every mole of phosphoric acid weighs 98 grams. The molar mass is calculated by summing the atomic masses of all the atoms in the molecule:

• Hydrogen (H): 1 gram/mole \(\times 3 = 3\) grams/mole
• Phosphorus (P): 31 grams/mole
• Oxygen (O): 16 grams/mole \(\times 4 = 64\) grams/mole

Adding these gives \(3 + 31 + 64 = 98\) grams/mole. This concept is crucial for converting between the mass of a reactant or product and the amount in moles.

Concentration Calculations

Concentration is typically expressed as a percentage or in terms of molarity. In this case, we are dealing with concentrated sulfuric acid that is 98% pure.
This means for every 100 grams of solution, 98 grams is pure sulfuric acid. The adjustment made for concentration involves dividing the mass needed by the percentage concentration, which converts the mass to represent what is needed in the actual chemical form: \[\text{Actual mass} = \frac{3000 \text{ kg}}{0.98}\] The equation adjusts the calculation to account for the solvent and provide the true amount of acid necessary for the reaction.

Density and Volume Calculations

Density is a property that relates mass to volume and is particularly useful in chemistry for converting between these two quantities.

• Density (\(d\)) is defined as mass (\(m\)) per unit volume (\(V\)) and expressed as \(d = \frac{m}{V}\).

In this scenario, the density of concentrated sulfuric acid is given as \(1.84\, \mathrm{g/mL}\). This allows for the conversion of the calculated mass of sulfuric acid (adjusted for concentration) into a volume, which is often more practical for measurements in both lab and industrial settings: \[\text{Volume} = \frac{3,061,220 \text{ g}}{1.84 \text{ g/mL}} = 1,663,706.52 \text{ mL}\]Converting this to liters involves simply dividing by 1,000 (since there are 1,000 milliliters in a liter), resulting in \(1,663.71 \text{ L}\), which is a volumetric measurement of how much concentrated sulfuric acid is needed for the reaction.