Question

Consider the balanced chemical equation \\[ A+5 B \rightarrow 3 C+4 D \\] When equal masses of \(A\) and \(B\) are reacted, which is limiting, A or B? Justify your choice. a. If the molar mass of \(A\) is greater than the molar mass of \(\mathrm{B}\), then \(\mathrm{A}\) must be limiting. b. If the molar mass of \(\mathrm{A}\) is less than the molar mass of \(\mathrm{B}\), then \(\mathrm{A}\) must be limiting. c. If the molar mass of \(A\) is greater than the molar mass of \(\mathrm{B}\), then \(\mathrm{B}\) must be limiting. d. If the molar mass of \(A\) is less than the molar mass of \(\mathrm{B},\) then \(\mathrm{B}\) must be limiting.

Short answer

a. If the molar mass of $A$ is greater than the molar mass of $B$, then $A$ must be limiting. d. If the molar mass of $A$ is less than the molar mass of $B$, then $B$ must be limiting.

Step-by-step solution

Determine the molar ratios of reactants

The balanced chemical equation is \[ A + 5 B \rightarrow 3 C + 4 D \] According to the equation, 1 mole of A reacts with 5 moles of B.

Determine the mass ratios of reactants

Since we are given that equal masses of A and B react, let the mass of A be mA and the mass of B be mB, with mA = mB. We will now relate their molar masses to the masses: Moles of A = \(\frac{mA}{M_A}\) Moles of B = \(\frac{mB}{M_B}\)

Compare the moles of reactants

As we have determined the moles of A and B, we will now compare them to the stoichiometric ratio: Ratio of moles (A to B) = \(\frac{\text{Moles of A}}{\text{Moles of B}}\) = \(\frac{\frac{mA}{M_A}}{\frac{mB}{M_B}}\) = \(\frac{M_B}{M_A}\) Because mA = mB, the ratio of their moles depends on the ratio of their molar masses. Recall that in the balanced chemical equation, the ratio of moles should be \(\frac{1}{5}\) for a complete reaction.

Analyze the given options

We will now analyze the provided options and compare them with our finding from Step 3: a. If the molar mass of A is greater than the molar mass of B, then A must be limiting. (Ratio = \(\frac{M_B}{M_A}\) < 1) b. If the molar mass of A is less than the molar mass of B, then A must be limiting. (Ratio = \(\frac{M_B}{M_A}\) > 1) c. If the molar mass of A is greater than the molar mass of B, then B must be limiting. (Ratio = \(\frac{M_B}{M_A}\) < 1) d. If the molar mass of A is less than the molar mass of B, then B must be limiting. (Ratio = \(\frac{M_B}{M_A}\) > 1)

Conclude which reactant is limiting

Comparing the options with the desired stoichiometric ratio of \(\frac{1}{5}\): Option a: If the molar mass of A is greater than the molar mass of B, the ratio (\(\frac{M_B}{M_A}\)) will be less than 1 which is less than the stoichiometric ratio. Therefore, A will be limiting because it has fewer moles compared to the required ratio. Option b: If the molar mass of A is less than the molar mass of B, the ratio (\(\frac{M_B}{M_A}\)) will be greater than 1 which is greater than the stoichiometric ratio. Therefore, B will be limiting because it has fewer moles compared to the required ratio. Therefore, the correct answer is a combination of options: A limiting: If the molar mass of A is greater than the molar mass of B. B limiting: If the molar mass of A is less than the molar mass of B.

Key concepts

Chemical Equation Balancing

Understanding how to balance chemical equations is a foundational skill in chemistry. When we balance a chemical equation, we adjust the coefficients in front of reactants and products so that the number of atoms for each element is the same on both sides of the equation.

Take the equation from the exercise:
\[ A + 5B \rightarrow 3C + 4D \]
Here, the equation shows that 1 molecule of A reacts with 5 molecules of B to produce 3 molecules of C and 4 molecules of D. Balancing chemical equations ensures the law of conservation of mass is upheld, which states that matter cannot be created or destroyed in a chemical reaction.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, between reactants and products in chemical reactions. It allows chemistry students to predict the amounts of substances consumed and produced in a given reaction.

For example, stoichiometry can tell us that if we start with equal masses of A and B in our exercise, knowing their molar masses can help us calculate which one will limit the reaction. By using the molar ratios provided by the balanced equation, we can determine that for every mole of A, we need 5 moles of B to react completely.

Molar Mass

The molar mass of a substance is the mass of one mole of that substance, which is numerically equivalent to the average atomic mass of the element or compound expressed in grams. The molar mass of each element is found on the periodic table and is vital for converting between grams and moles.

In our exercise solution, we look at the molar masses of substances A and B. This tells us that if substance A has a larger molar mass than substance B, and we start with equal masses of each, there will be fewer moles of A than B. As a result, A would be the limiting reactant since lesser moles of A are available for the reaction with B.

Mole-to-Mole Ratio

The mole-to-mole ratio is taken directly from the coefficients of a balanced chemical equation. It describes how many moles of one substance react or form in relation to another substance in the reaction.

Looking at our equation \( A + 5B \rightarrow 3C + 4D \), we see that the mole-to-mole ratio between A and B is 1:5. This means that 1 mole of A is required to react with 5 moles of B. When considering the relative amounts of each reactant we have, this ratio becomes crucial in determining the limiting reactant, which is the reactant that is completely consumed first and thus limits the extent of the reaction.