Question

Consider the reaction between silver and nitric acid for which the unbalanced equation is $$ \mathrm{Ag}(s)+\mathrm{H}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q) \longrightarrow \mathrm{Ag}^{+}(a q)+\mathrm{NO}_{2}(g)+\mathrm{H}_{2} \mathrm{O} $$ (a) Balance the equation. (b) If \(42.50 \mathrm{~mL}\) of \(12.0 \mathrm{M}\) nitric acid furnishes enough \(\mathrm{H}^{+}\) to react with silver, how many grams of silver react?

Short answer

Answer: 41.24 g of silver reacts with the given nitric acid solution.

Step-by-step solution

Balance the chemical equation

To balance the chemical equation, we need to make sure that the number of atoms of each element remains the same in the reactants and products. The balanced chemical equation is: $$ 3 \mathrm{Ag}(s) + 4 \mathrm{H}^{+}(aq) + 4 \mathrm{NO}_{3}^{-}(aq) \longrightarrow 3 \mathrm{Ag}^{+}(aq) + 2 \mathrm{NO}_{2}(g) + 2 \mathrm{H}_{2}\mathrm{O}(l) $$

Find the moles of nitric acid

We are given that the volume of nitric acid is \(42.50 \mathrm{~mL}\) and its molarity is \(12.0 \mathrm{M}\), which means there are \(12.0\) moles of nitric acid in \(1\) liter of the solution. To find the moles of nitric acid, we use the formula: moles of solute = molarity \(\times\) volume (in liters). So, moles of nitric acid \(= 12.0 \mathrm{M} \times 42.50 \times 10^{-3} \mathrm{L} = 0.510 \mathrm{~mol}\).

Find the moles of silver that reacted

Using the balanced chemical equation, we can see that \(4\) moles of nitric acid react with \(3\) moles of silver. Using the stoichiometry, we can now set up the ratio: $$ \frac{3 \mathrm{~moles~Ag}}{4 \mathrm{~moles~HNO_3}} = \frac{x \mathrm{~moles~Ag}}{0.510 \mathrm{~moles~HNO_3}} $$ Solving for \(x\) moles of silver: $$ x = \frac{3\cdot 0.510}{4} = 0.3825 \mathrm{~moles~Ag} $$

Convert moles of silver to mass of silver

To find the mass of silver that reacted, we multiply the number of moles of silver found in Step 3 by the molar mass of silver: $$ \text{mass of silver} = 0.3825 \mathrm{~moles~Ag} \times 107.87 \frac{\mathrm{g}}{\mathrm{mol}} = 41.24 \mathrm{~g} $$ Therefore, \(41.24 \mathrm{~g}\) of silver react with the given nitric acid.

Key concepts

Chemical Equation Balancing

Balancing a chemical equation is crucial for accurately representing a chemical reaction. Each side of the equation must have the same number of atoms for each element involved. It's like a balancing game with numbers. Here’s how you do it.

- **Identify each atom**: First, list all the atoms in the reactants and products. In the original equation, we have silver (Ag), hydrogen (H), nitric oxide ions (NO₃), and water (H₂O).
- **Count and Compare**: Count how many atoms of each element appear on both sides. The goal is to have an equal count on both sides of the equation.
- **Adjust coefficients**: Use coefficients (the numbers before molecules) to balance each type of atom. For example, we balanced 3 Ag with 3 Ag⁺, and adjusted coefficients to balance hydrogen and nitrogen in the reactants and products.
- **Check your work**: Verify that all atom counts match on each side of the equation after adding coefficients. For this reaction, correcting it gives: \[ 3 \mathrm{Ag}(s) + 4 \mathrm{H}^+(aq) + 4 \mathrm{NO}_3^-(aq) \rightarrow 3 \mathrm{Ag}^+(aq) + 2 \mathrm{NO}_2(g) + 2 \mathrm{H}_2\mathrm{O}(l) \]
Balancing helps to ensure that the conservation of mass principle is observed and the reaction is correctly understood.

Molarity Calculation

Molarity is a way to express the concentration of a solution, representing the number of moles of solute per liter of solution. For calculating molarity, there’s a straightforward formula: molarity \( M = \frac{\text{moles of solute}}{\text{liters of solution}} \).

When given a problem, follow these steps:
- **Measure volume in liters**: Convert the given volume from milliliters to liters by dividing by 1000. For instance, 42.50 mL is 0.0425 liters.
- **Use the molarity formula**: Multiply the molarity by the volume in liters. In our case, 12.0 M \( \times 0.0425 \text{ L} = 0.510 \text{ moles} \).
This calculation helps in figuring out how much of a chemical will react, based on its molarity. Understanding this concept allows you to engage meaningfully with a wide range of chemical reactions and experiments.

Moles to Mass Conversion

Converting moles to mass is a common step in stoichiometry and involves using the molar mass of elements. The molar mass acts as a bridge between the number of moles and the grams of a substance.

Here’s how to perform this conversion:
- **Identify the molar mass**: Determine the molar mass of the element or compound, usually given in grams per mole (g/mol). For silver (Ag), it is approximately 107.87 g/mol.
- **Calculate the mass**: Multiply the number of moles by the molar mass. From our example, using 0.3825 moles of Ag:
\[ \text{mass} = 0.3825 \text{ moles} \times 107.87 \frac{\text{g}}{\text{mol}} = 41.24 \text{ grams} \]
This conversion is essential in determining how much of a substance is involved in a reaction in terms of weight, which can be very practical in laboratory settings.