Question

Consider the balanced equation $$ \mathrm{C}_{3} \mathrm{H}_{3}(g)+5 \mathrm{O}_{2}(g) \rightarrow 3 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g) $$ What mole ratio enables you to calculate the number of moles of oxygen needed to react exactly with a given number of molcs of \(\mathrm{C}_{3} \mathrm{H}_{3}(g)\) ? What mole ratios enable you to calculate how many moles of each product form from a given number of moles of \(\mathrm{C}_{3} \mathrm{H}_{8}\) ?

Short answer

The mole ratios from the balanced equation \(\mathrm{C}_{3}\mathrm{H}_{3}(g)+5\mathrm{O}_{2}(g) \rightarrow 3\mathrm{CO}_{2}(g)+4\mathrm{H}_{2}\mathrm{O}(g)\) are as follows: - 5 moles of O₂ are needed to react exactly with 1 mole of C₃H₃: Mole ratio \(= \frac{5}{1} = 5\). - 3 moles of CO₂ are formed for every 1 mole of C₃H₃: Mole ratio \(= \frac{3}{1} = 3\). - 4 moles of H₂O are formed for every 1 mole of C₃H₃: Mole ratio \(= \frac{4}{1} = 4\).

Step-by-step solution

Identify the stoichiometry of the balanced equation

The balanced chemical equation is: $$ \mathrm{C}_{3} \mathrm{H}_{3}(g)+5 \mathrm{O}_{2}(g) \rightarrow 3 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g) $$ The coefficients in the balanced equation represent the moles of each species involved in the reaction. From the balanced equation, we have: - 1 mol of C₃H₃ reacts with 5 mol of O₂ to produce 3 mol of CO₂ and 4 mol of H₂O. Now we can find the mole ratios.

Calculate the mole ratio for O₂

To calculate the number of moles of oxygen needed to react with a given number of moles of C₃H₃, use the mole ratio of O₂ to C₃H₃ from the balanced equation: Mole ratio \(= \frac{\text{moles of O}_2}{\text{moles of C}_{3}\text{H}_{3}}\) From the balanced equation, we have 1 mol C₃H₃ reacting with 5 mol O₂: Mole ratio \(= \frac{5}{1} = 5\) Thus, 5 moles of O₂ are needed to react exactly with 1 mole of C₃H₃.

Calculate the mole ratios for CO₂ and H₂O

To calculate the number of moles of each product formed from a given number of moles of C₃H₈, use the mole ratios of CO₂ and H₂O to C₃H₈ from the balanced equation: Mole ratios: - \(= \frac{\text{moles of CO}_{2}}{\text{moles of C}_{3}\text{H}_{3}}\) - \(= \frac{\text{moles of H}_{2}\text{O}}{\text{moles of C}_{3}\text{H}_{3}}\) From the balanced equation, we have: - 1 mol C₃H₃ produces 3 mol CO₂ - 1 mol C₃H₃ produces 4 mol H₂O Mole ratios for the products: - \(= \frac{3}{1} = 3\) for CO₂ - \(= \frac{4}{1} = 4\) for H₂O Thus, from a given number of moles of C₃H₈: - 3 times as many moles of CO₂ will be formed. - 4 times as many moles of H₂O will be formed.

Key concepts

Chemical Reaction Balancing

Balancing a chemical reaction is a fundamental skill in chemistry, often described as the art of ensuring that the number of atoms for each element is the same on both the reactants and the products side of a chemical equation. When we examine the reaction \[\mathrm{C}_{3} \mathrm{H}_{3}(g)+5 \mathrm{O}_{2}(g) \rightarrow 3 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)\]we are looking at a balanced chemical equation. This means that the carbon and hydrogen atoms from the reactant \(\mathrm{C}_{3} \mathrm{H}_{3} \) are conserved and are transformed into carbon dioxide (CO₂) and water (H₂O) molecules in the same quantity as they appeared in the reactant molecule. The oxygen atoms are also balanced, with five oxygen molecules (consisting of two atoms each) reacting to form three CO₂ molecules and four H₂O molecules, each containing oxygen atoms.

The coefficients before each compound (for example, the '5' before \(\mathrm{O}_{2} \) and the '3' before CO₂) indicate the exact number of moles of each substance that take part in the reaction. This is critical for understanding the quantitative aspects of a reaction, which is where stoichiometry comes into play.

Mole Ratio Calculation

The mole ratio is an essential concept in stoichiometry as it tells us how much of one substance will react or form in relation to another substance in a chemical reaction. In the provided exercise, the balanced chemical equation indicates that to completely react one mole of \(\mathrm{C}_{3} \mathrm{H}_{3} \) gas, you would need five moles of oxygen gas (\(\mathrm{O}_{2} \)), because the mole ratio of \(\mathrm{O}_{2} \) to \(\mathrm{C}_{3} \mathrm{H}_{3} \) is 5:1. This implies that if you have, say, 2 moles of \(\mathrm{C}_{3} \mathrm{H}_{3} \), you would need \(2 \times 5 = 10 \) moles of \(\mathrm{O}_{2} \) for complete reaction.

The mole ratio can be used inversely as well—if you know the amount of oxygen available, you can determine how much \(\mathrm{C}_{3} \mathrm{H}_{3} \) can be completely reacted. This concept empowers you to calculate reactants and products accurately, which is incredibly important in lab settings and industrial applications where precise measurements can affect the outcome of a reaction.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers written in front of the reactants and products in a chemical equation. They are fundamental to stoichiometry because they indicate the proportion in which the substances react. These coefficients represent the moles of each substance and thus, they guide the calculation of the mole ratios. In our exercise, the stoichiometric coefficients are '1' for \(\mathrm{C}_{3} \mathrm{H}_{3} \), '5' for \(\mathrm{O}_{2} \), '3' for CO₂, and '4' for H₂O.

Knowing the stoichiometric coefficients helps us figure out how much product forms from a given reactant or how much of a reactant is required to make a certain amount of product. If we take the coefficient of '1' for \(\mathrm{C}_{3} \mathrm{H}_{3} \), and compare it with the '3' for CO₂ and the '4' for H₂O, we can quickly conclude that one mole of \(\mathrm{C}_{3} \mathrm{H}_{3} \) will produce three moles of CO₂ and four moles of H₂O. These proportional relationships are vital for chemists to scale reactions up or down and to calculate yields, which is the efficiency of a chemical reaction.