Mole ratios Knowledge-Based Study

Study notes

1. Understanding Mole Ratios

   In chemistry, a mole ratio is a conversion factor that relates the amounts in moles of any two substances involved in a chemical reaction. These ratios are derived directly from the numerical coefficients of a balanced chemical equation.

   Consider the synthesis of ammonia via the Haber process:

   $$\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$$

   The stoichiometric coefficients tell us the exact relative number of particles—and therefore the exact relative number of moles—that react and form:

   • 1 mole of $\text{N}_2$ reacts with 3 moles of $\text{H}_2$.
   • 1 mole of $\text{N}_2$ produces 2 moles of $\text{NH}_3$.
   • 3 moles of $\text{H}_2$ produce 2 moles of $\text{NH}_3$.

   Constructing Conversion Factors

   Any balanced chemical equation provides multiple conversion factors depending on which two species are being compared. For any two substances $A$ and $B$, the mole ratio can be written as:

   $$\frac{\text{moles of } A}{\text{moles of } B} \quad \text{or} \quad \frac{\text{moles of } B}{\text{moles of } A}$$

   Using the formation of water as an example:

   $$2\text{H}_2(g) + \text{O}_2(g) \rightarrow 2\text{H}_2\text{O}(l)$$

   The possible mole ratios between hydrogen gas, oxygen gas, and water include:

   • Between reactants: $\frac{2\text{ mol H}_2}{1\text{ mol O}_2}$ or $\frac{1\text{ mol O}_2}{2\text{ mol H}_2}$
   • Between a reactant and product: $\frac{2\text{ mol H}_2\text{O}}{1\text{ mol O}_2}$ or $\frac{2\text{ mol H}_2}{2\text{ mol H}_2\text{O}}$ (which simplifies to $1:1$)

   The choice of which substance is placed in the numerator depends on which unit you need to cancel during dimensional analysis.

   Calculating Unknown Quantities

   Mole ratios act as the bridge that allows you to calculate the moles of an unknown substance from the known moles of another substance in a reaction.

   General Formula

   $$\text{Moles of Unknown} = \text{Moles of Known} \times \left( \frac{\text{Coefficient of Unknown}}{\text{Coefficient of Known}} \right)$$

   Worked Example

   How many moles of $\text{O}_2$ are required to react completely with 4.5 moles of propane ($\text{C}_3\text{H}_8$)?

   $$\text{C}_3\text{H}_8(g) + 5\text{O}_2(g) \rightarrow 3\text{CO}_2(g) + 4\text{H}_2\text{O}(g)$$

   1. Identify the known and unknown: Known = $4.5\text{ mol C}_3\text{H}_8$; Unknown = moles of $\text{O}_2$.
   2. Identify the mole ratio: According to the balanced equation, $1\text{ mol C}_3\text{H}_8$ reacts with $5\text{ mol O}_2$, giving the ratio $\frac{5\text{ mol O}_2}{1\text{ mol C}_3\text{H}_8}$.
   3. Perform the calculation:
      $$4.5\text{ mol C}_3\text{H}_8 \times \frac{5\text{ mol O}_2}{1\text{ mol C}_3\text{H}_8} = 22.5\text{ mol O}_2$$

   Essential Rules for Success

   • Always balance the equation first: Mole ratios obtained from an unbalanced equation will always lead to incorrect stoichiometric results.
   • Moles, not grams: Coefficients represent ratios of particles and moles, never mass ratios. For example, in $2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O}$, 2 grams of $\text{H}_2$ do not react with 1 gram of $\text{O}_2$. You must convert masses to moles before using mole ratios.
   • Dimensional consistency: Set up your dimensional analysis so that the unit of the given substance cancels out, leaving only the unit of the target substance.