Question

Consider the reaction: $$\mathrm{Li}_{2} \mathrm{S}(a q)+\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}(a q) \longrightarrow 2 \mathrm{LiNO}_{3}(a q)+\mathrm{CoS}(s)$$ What volume of 0.150 \(\mathrm{M} \mathrm{Li}_{2} \mathrm{S}\) solution is required to completely react with 125 \(\mathrm{mL}\) of 0.150 \(\mathrm{M} \mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}\) ?

Short answer

To completely react with 125 mL of 0.150 M \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\), 125 mL of 0.150 M \(\mathrm{Li}_{2}S\) solution is required.

Step-by-step solution

Write the balanced chemical equation

First, confirm the chemical equation is balanced. The given equation \(\mathrm{Li}_{2}S (aq) + \mathrm{Co}(\mathrm{NO}_{3})_{2} (aq) \rightarrow 2\mathrm{LiNO}_{3} (aq) + \mathrm{CoS} (s)\) is already balanced, showing a 1:1 molar ratio between \(\mathrm{Li}_{2}S\) and \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\).

Identify the molar relationship

The balanced equation indicates that one mole of \(\mathrm{Li}_{2}S\) reacts with one mole of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\). This 1:1 molar ratio will be used to determine the required volume of \(\mathrm{Li}_{2}S\).

Calculate moles of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\)

Use the concentration and volume of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\) to find the moles. Moles of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\) are given by the formula: \(\text{moles} = \text{concentration} \times \text{volume}\). Therefore, \(0.150 \text{ M} \times 0.125 \text{ L} = 0.01875 \text{ moles}\) of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\) are present.

Calculate the required moles of \(\mathrm{Li}_{2}S\)

Using the 1:1 molar ratio from the balanced equation, we need 0.01875 moles of \(\mathrm{Li}_{2}S\) to react with the 0.01875 moles of \(\mathrm{Co}(\mathrm{NO}_{3})_{2}\).

Calculate the required volume of \(\mathrm{Li}_{2}S\) solution

To find the required volume of \(\mathrm{Li}_{2}S\), use the formula: \(\text{volume} = \frac{\text{moles}}{\text{concentration}}\). Therefore, the required volume of \(\mathrm{Li}_{2}S\) is \(\frac{0.01875 \text{ moles}}{0.150 \text{ M}} = 0.125 \text{ L}\) or 125 mL.

Key concepts

Chemical Equations

A chemical equation is a symbolic representation of a chemical reaction, where reactants are shown on the left side and products on the right side. Each compound is denoted by its chemical formula, and the numbers in front of these formulas are called coefficients. The coefficients indicate the number of molecules or moles required for the reaction to occur under stoichiometric conditions—meaning the quantities are in perfect proportions to use up all reactants without leftovers. In our exercise, we reviewed the equation
\[ \mathrm{Li}_2\mathrm{S}(aq) + \mathrm{Co}(\mathrm{NO}_3)_2(aq) \longrightarrow 2\mathrm{LiNO}_3(aq) + \mathrm{CoS}(s) \]
It illustrates the reaction between aqueous lithium sulfide (\(\mathrm{Li}_2\mathrm{S}\)) and aqueous cobalt(II) nitrate (\(\mathrm{Co}(\mathrm{NO}_3)_2\)) to form aqueous lithium nitrate (\(\mathrm{LiNO}_3\)) and solid cobalt(II) sulfide (\(\mathrm{CoS}\)). It is crucial for the chemical equation to be balanced, which in this case it already is, with equal numbers of each type of atom on both sides of the equation.

Molar Ratio

The molar ratio is a key concept in stoichiometry that tells us the proportion of reactants that react with each other in a chemical equation. This ratio is derived from the coefficients of the balanced chemical equation and informs us about the relative amounts of reactants needed for the reaction. In the given exercise, the balanced equation indicates a 1:1 molar ratio between \(\mathrm{Li}_2\mathrm{S}\) and \(\mathrm{Co}(\mathrm{NO}_3)_2\). This means that one mole of \(\mathrm{Li}_2\mathrm{S}\) is required to react with one mole of \(\mathrm{Co}(\mathrm{NO}_3)_2\). Understanding this fundamental stoichiometric relationship allows us to calculate the amount of one reactant needed when given the amount of the other. The proper usage of the molar ratio is critical in solving many stoichiometry problems, such as determining the volume of a solution needed to react with another substance.

Solution Concentration

Solution concentration is a measure of the amount of solute present in a given quantity of solvent or solution. It is typically expressed in units of molarity (M), where one molar is one mole of solute per liter of solution. In our exercise, we see the term '0.150 M \(\mathrm{Li}_2\mathrm{S}\)' which tells us there are 0.150 moles of lithium sulfide in every liter of that solution. Similarly, the concentration of cobalt(II) nitrate is given as '0.150 M', indicating the same amount of moles per liter. The concept of concentration is vital when we need to calculate the volume of a solution, as seen in Step 5 of the problem-solving process. Knowing the molarity and the total moles of solute allows us to use the formula
\[ \text{volume} = \frac{\text{moles}}{\text{concentration}} \]
to find the volume of solution required for complete reaction. This step is essential when preparing for experiments or when performing titrations in analytical chemistry.