Question

If \(0.50\) mole of \(\mathrm{BaCl}_{2}\) is mixed with \(0.20\) mole of \(\mathrm{Na}_{3} \mathrm{PO}_{4}\), the maximum number of moles of \(\mathrm{Ba}_{3}\left(\mathrm{PO}_{4}\right)_{2}\) that can be formed is: (a) \(0.10\) (b) \(0.20\) (c) \(0.30\) (d) \(0.40\)

Short answer

The maximum number of moles of \(\text{Ba}_3\text{(PO}_4\text{)}_2\) formed is 0.10. Answer: (a) 0.10.

Step-by-step solution

Write the Balanced Equation

The reaction between barium chloride \(\text{(BaCl}_2\text{)}\) and sodium phosphate \(\text{(Na}_3\text{PO}_4\text{)}\) produces barium phosphate \(\text{(Ba}_3\text{(PO}_4\text{)}_2\text{)}\) and sodium chloride \(\text{(NaCl)}\). The balanced chemical equation is: \[ 3\text{BaCl}_2 + 2\text{Na}_3\text{PO}_4 \rightarrow \text{Ba}_3\text{(PO}_4\text{)}_2 + 6\text{NaCl} \] This equation will guide the stoichiometry calculations.

Determine the Limiting Reagent

From the balanced equation, 3 moles of \(\text{BaCl}_2\) react with 2 moles of \(\text{Na}_3\text{PO}_4\). Thus, the mole ratio is \(\frac{3}{2}\). The moles of \(\text{BaCl}_2\) provided are 0.50 and the moles of \(\text{Na}_3\text{PO}_4\) provided are 0.20. Calculate how many moles of \(\text{BaCl}_2\) are required for 0.20 moles of \(\text{Na}_3\text{PO}_4\): \[ \text{Required moles of } \text{BaCl}_2 = 0.20 \times \frac{3}{2} = 0.30 \text{ moles} \] Since we only have 0.50 moles of \(\text{BaCl}_2\), compare the required and available moles, \(\text{Na}_3\text{PO}_4\) is the limiting reagent.

Calculate Maximum Moles of Product

Using the limiting reagent (\(\text{Na}_3\text{PO}_4\)), calculate the moles of \(\text{Ba}_3\text{(PO}_4\text{)}_2\) that can be formed. From the balanced equation, 2 moles of \(\text{Na}_3\text{PO}_4\) produce 1 mole of \(\text{Ba}_3\text{(PO}_4\text{)}_2\). Thus, \[ \text{Moles of } \text{Ba}_3\text{(PO}_4\text{)}_2 = 0.20 \times \frac{1}{2} = 0.10 \text{ moles} \] The maximum number of moles of \(\text{Ba}_3\text{(PO}_4\text{)}_2\) that can be formed is 0.10.

Key concepts

Limiting Reagent

In every chemical reaction, the reactant that is completely used up first is known as the limiting reagent. It determines the maximum amount of product that can be formed, as once the limiting reagent is consumed, the reaction cannot proceed further. Identifying the limiting reagent is a crucial task in stoichiometry.
To find out which reactant is the limiting reagent, one must compare the mole ratios of the reactants given in the problem to those in the balanced chemical equation. In this exercise, we compared the given moles of barium chloride ( 0.50 ) and sodium phosphate ( 0.20 ) with the stoichiometric ratios from the equation (3:2) .
By calculating how much barium chloride is needed for 0.20 moles of sodium phosphate, we confirmed that we require 0.30 moles of barium chloride. As there were more than enough moles of barium chloride available, sodium phosphate became the limiting reagent. Knowing your limiting reagent will help you determine the amount of product formed.

Balanced Chemical Equation

A balanced chemical equation is vital for understanding and solving stoichiometry problems. It ensures that the number of atoms for each element is equal on both sides of the equation, reflecting the law of conservation of mass. This balance indicates the specific proportions or moles in which reactants combine to form a product.
In this exercise, the equation is: 3BaCl_2 + 2Na_3PO_4 → Ba_3(PO_4)_2 + 6NaCl. This means 3 moles of barium chloride react with 2 moles of sodium phosphate to produce 1 mole of barium phosphate and 6 moles of sodium chloride. Once you have this balance, it forms the foundation for all stoichiometric calculations in the reaction.
Balancing equations involves adjusting coefficients to ensure equivalent numbers of each type of atom on both sides of the reaction arrow. It is an essential skill for predicting the amounts of reactants and products involved in a chemical reaction.

Mole Ratio

The concept of mole ratio is central to solving stoichiometry problems, as it tells you how much of one substance will react with or produce another. Derived from the coefficients in the balanced equation, mole ratios help translate moles of one reactant or product into moles of another.
In our example, the balanced reaction equation shows that 3 moles of barium chloride react with 2 moles of sodium phosphate. This results in a mole ratio of 3:2. Similarly, 2 moles of sodium phosphate will yield 1 mole of barium phosphate, indicating a 2:1 mole ratio.
To determine the amount of product that can be formed or how much of each reactant is needed, you multiply or divide by these ratios. In our exercise, using the mole ratio, we found that 0.20 moles of sodium phosphate would form 0.10 moles of barium phosphate, showcasing the practical application of the mole ratio in calculations.