Question

A mixture of A and B contains a total of 5.3 mols. Both A and B react with Z according to the following equations: $$ \begin{array}{c} \mathrm{A}+\mathrm{Z} \longrightarrow \mathrm{AZ} \\ \mathrm{B}+2 \mathrm{Z} \longrightarrow \mathrm{BZ}_{2} \end{array} $$ The reaction of the mixture of A and B with Z consumes \(7.8 \mathrm{~mol} \mathrm{Z}\). Assuming the reactions go to completion, how many moles of A does the mixture contain?

Short answer

The mixture contains 2.8 moles of A.

Step-by-step solution

Assign Variables

Let the number of moles of A be represented by x, and the number of moles of B be represented by y. We now have two variables to represent the amounts of A and B.

Set Up the First Equation

The problem states that the total moles of A and B equals 5.3. This gives us our first equation: \( x + y = 5.3 \).

Set Up the Second Equation

The problem also states that the reaction with Z consumes a total of 7.8 moles of Z. Since one mole of A reacts with one mole of Z and one mole of B reacts with two moles of Z, we get our second equation: \( x + 2y = 7.8 \).

Solve the System of Equations

Now we have a system of linear equations:\[\begin{align*}x + y &= 5.3 \ x + 2y &= 7.8\end{align*}\]We can use substitution or elimination to solve for x and y. Let's opt for the elimination method by subtracting the first equation from the second to find the value of y.

Subtract the First Equation from the Second

Subtracting the first equation from the second one to eliminate x gives us:\[ (x + 2y) - (x + y) = 7.8 - 5.3 \]which simplifies to \[ y = 2.5 \].

Solve for x

With the value of y known, we substitute y = 2.5 into the first equation \( x + y = 5.3 \) to find x: \[ x + 2.5 = 5.3 \] which simplifies to \[ x = 5.3 - 2.5 = 2.8 \].

Key concepts

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry is crucial for understanding how substances interact in a chemical reaction. In the stoichiometry of a reaction, the quantities of reactants and products are expressed in moles, which allows chemists to predict the amounts of substances consumed and produced.

Let's take the example provided. We have reactants A and B reacting with Z to produce AZ and BZ₂ respectively. The stoichiometric coefficients, or the numbers in front of the substances in a chemical equation, indicate the proportions in which the reactants combine and the products form. In the equations given, A reacts with Z in a 1:1 ratio, whereas B reacts with 2 moles of Z for every mole of B. Understanding these ratios is key to solving the stoichiometry problem presented.

In the exercise, we calculate the moles of A (x) present in the mixture by analyzing the total reaction with Z. We then utilize the information about the total moles consumed to establish a relationship between A and B with Z, reflecting on the stoichiometric ratios provided by the balanced reactions.

Mole Concept

The mole concept is a bridge between the atomic world and the laboratory measurements. A mole is defined as the amount of substance that contains as many entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of pure carbon-12. This number is approximately 6.022 x 10²³, also known as Avogadro's number.

In our exercise, we deal with the mole concept by starting with a total of 5.3 moles of a mixture of A and B. When working with moles, it's important to understand that a mole represents a standardized unit that helps to calculate the mass of a substance needed or produced. The use of moles allows us to equate the mass of substance A to substance B, as long as the chemical equation is balanced, helping us to solve stoichiometry problems by equating the amount of reactants to the amount of products.

System of Linear Equations

A system of linear equations consists of two or more equations with two or more variables that can be graphically represented as straight lines on a coordinate plane. These equations are called 'linear' because they form a straight line when graphed. In the context of stoichiometry problems, the coefficients of the substances in the balanced chemical equations can be used to set up a system of linear equations.

In the provided example, the equations are set up based on the stoichiometric relationships between the reactants. The number of moles of A and B, represented by x and y, form a relationship that is expressed in two linear equations. We employ methods like substitution or elimination to solve this system for the unknown quantities. The elimination method in the step by step solution successfully isolated and determined the value of y, and subsequently x, showing how we can derive the quantities of each reactant present in a mixture.