Question

Consider the equation \(2 A+B \longrightarrow A_{2} B .\) If you mix \(1.0 \mathrm{~mole}\) of \(A\) with \(1.0\) mole of \(B\), what amount (moles) of \(A_{2} B\) can be produced?

Short answer

In the chemical reaction \(2A + B \longrightarrow A_{2}B\), when 1.0 mole of A and 1.0 mole of B are mixed, 0.5 moles of \(A_{2}B\) can be produced.

Step-by-step solution

Write the balanced chemical equation

\ First, let's write out the balanced chemical equation for the given reaction: \(2A + B \longrightarrow A_{2}B\) Since the equation is already balanced, we do not need to balance it.

Calculate the initial moles of reactants

\ The initial amounts of reactants A and B are given as: Moles of A = 1.0 mole Moles of B = 1.0 mole

Determine the limiting reactant

\ To find the limiting reactant, we must compare the mole ratio of A to B in the reaction. According to the balanced equation, 2 moles of A are required to react with 1 mole of B to produce A₂B. Divide the moles of each reactant by their respective coefficients from the balanced equation to find the reaction ratio. Reaction ratio A = Moles of A / 2 = 1.0 / 2 = 0.5 Reaction ratio B = Moles of B / 1 = 1.0 / 1 = 1.0 Since the reaction ratio of A is less than that of B, A is the limiting reactant in this reaction.

Calculate the moles of product A₂B produced

\ Now that we know A is the limiting reactant, we can use stoichiometry to determine the amount of product A₂B that can be formed. From the balanced equation, 2 moles of A will react with 1 mole of B to produce 1 mole of A₂B. Using the stoichiometric coefficients, we can set up the following proportion: (1 mole of A₂B / 2 moles of A) = (x moles of A₂B / 1.0 mole of A) Now, we can solve for x, which is the amount of product A₂B produced: x moles of A₂B = (1 mole of A₂B / 2 moles of A) * 1.0 mole of A x moles of A₂B = 0.5 mole of A₂B So, 0.5 moles of A₂B can be produced when 1.0 mole of A is mixed with 1.0 mole of B.

Key concepts

Balanced Chemical Equation

In chemistry, understanding a balanced chemical equation is key to predicting the outcomes of reactions. A balanced equation accurately represents what happens during the reaction, ensuring that the same number of atoms for each element are present on both sides of the reaction. This means that no atoms are lost or gained; they are simply rearranged.

In the given exercise, the equation is:

• 2A + B \longrightarrow A_{2}B

Notice the equation is already balanced. It displays how 2 moles of reactant A combine with 1 mole of reactant B to form the product \(A_{2}B\). This equation reflects the principle of conservation of mass, which is a cornerstone in stoichiometry.

To identify a balanced equation:

• Count the number of atoms on each side of the equation for each element involved.
• Adjust the coefficients (not subscripts) to ensure both sides have the same quantity of each atom.

A balanced chemical equation is essential for understanding the stoichiometry of a reaction and doing further calculations involving mole ratios.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, that exist in chemical reactions based on the balanced chemical equations. Through stoichiometry, one can calculate the amounts of reactants and products that are involved in a reaction.

In our example, stoichiometry helps us determine the amount of product \(A_{2}B\) produced. By examining the stoichiometric coefficients in the balanced equation \(2A + B \longrightarrow A_{2}B\), and using the limiting reactant, we can set a proportion to determine how much of the product will form from known quantities of reactants.

To perform stoichiometric calculations:

• Identify the balanced equation and relevant coefficients.
• Establish a ratio using these coefficients.
• Apply the ratio to the given amounts of reactants to find the unknown product quantity.

For instance, in our exercise, knowing the limiting reactant was crucial for applying stoichiometry to calculate that 0.5 moles of \(A_{2}B\) could be produced from 1.0 mole of A.

Mole Ratio

The mole ratio is a pivotal concept in stoichiometry, representing the ratio between the amounts in moles of any two substances involved in a chemical reaction. It is derived from the coefficients of a balanced chemical equation.

In our case, the balanced equation, \(2A + B \longrightarrow A_{2}B\), gives us molar ratios:

• 2 moles of A : 1 mole of B
• 2 moles of A : 1 mole of \(A_{2}B\)
• 1 mole of B : 1 mole of \(A_{2}B\)

These ratios indicate how many moles of one substance react with or produce moles of another in the equation.

To find the limiting reactant and calculate product quantities, divide the moles of each reactant by their coefficients in the chemical equation. In our case:

• \(\frac{1.0 \text{ mole of A}}{2} = 0.5\)
• \(\frac{1.0 \text{ mole of B}}{1} = 1.0\)

Since A has the smaller ratio, it is the limiting reactant, and it determines the maximum amount of product \(A_{2}B\) that can be formed, which is 0.5 moles. Understanding mole ratios is essential for calculating how much product results from a chemical reaction.