Question

Sodium borohydride, \(\mathrm{NaBH}_{4},\) reduces many metal ions to the metal. (a) Write a balanced equation for the reaction of \(\mathrm{NaBH}_{4}\) with \(\mathrm{AgNO}_{3}\) in water to give silver metal, \(\mathrm{H}_{2}\) gas, boric acid, and sodium nitrate. (The chemistry of \(\mathrm{NaBH}_{4}\) is described on pages \(982-983 .\) ) (b) What mass of silver can be produced from \(575 \mathrm{mL}\) of \(0.011 \mathrm{M} \mathrm{AgNO}_{3}\) and \(13.0 \mathrm{g}\) of \(\mathrm{NaBH}_{4} ?\)

Short answer

0.6823 g of silver can be produced.

Step-by-step solution

Write the Unbalanced Equation

First, we need to identify the reactants and products in the chemical reaction. The reactants are \( \mathrm{NaBH}_4 \) and \( \mathrm{AgNO}_3 \), and the products are \( \mathrm{Ag} \) (silver metal), \( \mathrm{H}_2 \) gas, \( \mathrm{H}_3\mathrm{BO}_3 \) (boric acid), and \( \mathrm{NaNO}_3 \). The unbalanced chemical equation can be written as: \[ \mathrm{NaBH}_4 + \mathrm{AgNO}_3 \rightarrow \mathrm{Ag} + \mathrm{H}_2 + \mathrm{H}_3\mathrm{BO}_3 + \mathrm{NaNO}_3 \]

Balance the Chemical Equation

To balance the equation, ensure the same number of each type of atom appears on both sides of the reaction. A balanced equation is: \[ 4\mathrm{AgNO}_3 + \mathrm{NaBH}_4 + 2\mathrm{H}_2O \rightarrow 4\mathrm{Ag} + \mathrm{H}_3\mathrm{BO}_3 + 4\mathrm{NaNO}_3 + 4\mathrm{H}_2 \] This equation satisfies the conservation of mass with equal numbers of each atom on each side.

Calculate Moles of \( \mathrm{AgNO}_3 \)

Using the volume and molarity of \( \mathrm{AgNO}_3 \), calculate the number of moles: \[ \text{moles of } \mathrm{AgNO}_3 = 0.575 \mathrm{L} \times 0.011 \mathrm{mol/L} = 0.006325 \mathrm{mol} \] This is the amount of silver nitrate available to react.

Calculate Moles of \( \mathrm{NaBH}_4 \)

Determine the moles of \( \mathrm{NaBH}_4 \) using its molar mass \((37.83 \mathrm{g/mol})\): \[ \text{moles of } \mathrm{NaBH}_4 = \frac{13.0 \mathrm{g}}{37.83 \mathrm{g/mol}} = 0.3437 \mathrm{mol} \] This is the amount of sodium borohydride available to react.

Identify Limiting Reactant

From the balanced equation, 1 mole of \( \mathrm{NaBH}_4 \) reacts with 4 moles of \( \mathrm{AgNO}_3 \). Compare moles available: \( \frac{0.006325}{4} = 0.001581 \mathrm{mol} \) of \( \mathrm{NaBH}_4 \) is needed, which is less than available \( \mathrm{NaBH}_4 \). Thus, \( \mathrm{AgNO}_3 \) is the limiting reactant.

Calculate Mass of Silver Produced

From the stoichiometry, \( 4 \mathrm{AgNO}_3 \) produces 4 \( \mathrm{Ag} \). Therefore, 0.006325 moles of \( \mathrm{AgNO}_3 \) produces 0.006325 moles of \( \mathrm{Ag} \). Calculate the mass of \( \mathrm{Ag} \): \[ \text{Mass of } \mathrm{Ag} = 0.006325 \mathrm{mol} \times 107.87 \mathrm{g/mol} = 0.6823 \mathrm{g} \] This is the mass of silver that can be produced.

Key concepts

Balancing Chemical Equations

Balancing chemical equations is a crucial step in any chemical reaction problem. In essence, it's like solving a puzzle, ensuring that atoms on the reactant side are equal in number to those on the product side. This follows the conservation of mass principle, which states that matter cannot be created or destroyed in a chemical reaction.

To balance a chemical equation, first identify all reactants and products and write their chemical formulas on either side of the reaction arrow. Start by balancing elements that appear only once on each side and use coefficients to equalize the number of atoms for each element. Make sure every element's atom count matches on both sides by adjusting these coefficients systematically.

• Always check hydrogen and oxygen last as they are often in multiple compounds.
• Double-check each atom's count to ensure correctness.

For our specific exercise, the balanced reaction portrays equal numbers of silver, hydrogen, boron, and nitrate groups on each side, maintaining the conservation of mass.

Limiting Reactant

When a chemical reaction takes place, the limiting reactant is the reactant that is completely consumed first, preventing any further reaction. Identifying this reactant allows us to calculate the maximum amount of product that can be formed.

To find the limiting reactant, first calculate the number of moles for each reactant. Using the molar ratio from the balanced equation, determine how much of one reactant is needed to fully react with the other. Compare the actual mole amounts available: whichever reactant runs out first is the limiting reactant.

• Use the stoichiometric coefficients from the balanced equation to make these comparisons.
• Calculate the amount of each reactant required to completely react with the other, and identify the one that is insufficient.

In our example, the sodium borohydride is in excess, making silver nitrate the limiting reactant.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantitative relationships of the reactants and products in a chemical reaction. It allows chemists to predict how much of a substance will be produced in a reaction based on the amounts of the reactants.

With stoichiometry, you use the balanced chemical equation to find relationships among the amounts of reactants and products. Begin by calculating moles, using coefficients from the balanced equation to transform reactant moles into product moles.

• Use the balanced equation to find the ratio of reactants to products.
• Convert moles to grams if needed using molar mass.

For example, in the given problem, stoichiometry tells us that four moles of silver nitrate yield four moles of silver metal.

Molarity Calculations

Molarity is a way to express the concentration of a solute in a solution, representing the number of moles of solute per liter of solution. It's paramount for calculating how reactants in solution will interact in a chemical reaction.

To calculate molarity, divide the number of moles of solute by the volume of the solution in liters. This gives the concentration, and multiplying this by the solution volume can provide the number of moles.

• Molarity (M) = Moles of solute / Volume of solution in liters.
• Convert any volume from milliliters to liters before calculations.

In our case, starting with 575 mL of a 0.011 M solution of silver nitrate provides us with 0.006325 moles of silver nitrate, which informs us of the reaction's stoichiometric calculations.