Question

Consider the equation \(2 \mathrm{A}+\mathrm{B} \rightarrow \mathrm{A}_{2} \mathrm{B}\). If you mix 1.0 mole of \(A\) and 1.0 mole of \(B,\) how many moles of \(\mathrm{A}_{2} \mathrm{B}\) can be produced?

Short answer

When 1.0 mole of A and 1.0 mole of B are mixed in the reaction \(2A + B \rightarrow A_2B\), the limiting reactant is A, and 0.25 moles of \(A_2B\) can be produced.

Step-by-step solution

Identify the chemical equation given

The given chemical equation is \[2A + B \rightarrow A_2B\] Here, 1 molecule of substance B reacts with 2 molecules of substance A to produce 1 molecule of substance A2B.

Identify the initial mole amounts of reactants given

We are given 1.0 mole of A and 1.0 mole of B.

Identify the stoichiometric coefficients and calculate the mole ratio

In the balanced chemical equation, the stoichiometric coefficients are the coefficients in front of chemical formulae. Here, we have: - 2 for A: \(2A\) - 1 for B: \(B\) The mole ratio of A to B is 2:1, which means for the reaction to complete, we need 2 moles of A for every mole of B.

Determine the limiting reactant

Now, we need to determine which reactant will be consumed first, limiting the reaction. To do this, we'll compare the given amount of reactants to the required mole ratio. For A, we have 1.0 mole (given amount) and need 2 moles to react with 1 mole of B (mole ratio). So we divide the given amount with the coefficient: \(1.0 \div 2 = 0.5\) For B, we have 1.0 mole (given amount) and need 1 mole to react with 2 moles of A (mole ratio). So we divide the given amount with the coefficient: \(1.0 \div 1 = 1\) The reactant with the smaller value is the limiting reactant, which in this case is A.

Calculate the amount of product produced based on the limiting reactant

Since A is the limiting reactant, the amount of product (A2B) produced will be based on the stoichiometry of A in the balanced chemical equation. From the stoichiometry, we know that 2 moles of A react to produce 1 mole of A2B. So, to find the amount of A2B produced, we use the value we got for A and multiply it by the ratio of moles of A2B to A: \(0.5 \times \frac{1}{2} = 0.25\)

Summarize the result

The amount of A2B that can be produced in the reaction is 0.25 moles.

Key concepts

Limiting Reactant

In any chemical reaction, the limiting reactant is the substance that is completely consumed first, thus stopping the reaction from continuing. This reactant determines the maximum amount of product that can be formed.
To identify the limiting reactant, we compare the mole ratio of the reactants with their initial quantities. The reactant with the lesser ratio completion leads to it being the limiting reactant.
In the given equation, the limiting reactant is identified by evaluating the stoichiometric requirements of the reaction. For our example, since substance A requires 2 moles for every mole of B, we find that 1 mole of A is insufficient to fully react with 1 mole of B. Thus, A is the limiting reactant because it prevents more \(A_2B\) from being produced.
This step is crucial in stoichiometry as it directly impacts the calculation of the theoretical yield of the products.

Mole Ratio

The mole ratio is derived from the coefficients of the reactants and products in a balanced chemical equation. It helps determine the proportions in which substances react.

• A balanced equation tells us the exact ratio in which reactants combine and products form. For example, the ratio of A to B is 2:1 in our equation.
• This means 2 moles of A are needed for every 1 mole of B to form products.
• Mole ratios are instrumental in stoichiometry calculations, allowing us to extract the stoichiometric relationships between different substances in the reaction.

Understanding mole ratios is vital since it enables prediction of how much of one reactant will react with a given amount of another. Likewise, it helps in calculating how much product will form.

Chemical Equation

A chemical equation is a symbolic representation of a chemical reaction. It shows the reactants transforming into products, using their chemical formulas.
In a chemical equation, the reactants are written on the left side, while products are on the right, separated by an arrow indicating the reaction direction. For instance, in the given equation for forming \(A_2B\), reactants are 2 A and B, resulting in product A2B.
Chemical equations must be balanced to satisfy the law of conservation of mass, meaning the same number of atoms of each element should be present on both sides of the equation.
Balancing the equation requires adjusting the coefficients of each compound or element, which relate directly to the stoichiometric coefficients used in problems like this one.

Stoichiometric Coefficients

Stoichiometric coefficients are numerical coefficients in a chemical equation representing the number of moles of each substance involved in the reaction.
They are crucial for balancing chemical reactions and determining the amount of reactants and products.

• For example, in the equation \(2A + B ightarrow A_2B\), the stoichiometric coefficients are 2 for A, 1 for B, and 1 for \(A_2B\).
• These coefficients provide the necessary information to calculate mole ratios and identify limiting reactants.

In practice, stoichiometric coefficients help predict how much product can be formed from given quantities of reactants and are an essential part of solving stoichiometry problems, ensuring precise calculations in chemical reactions.