Question

An ionic compound \(\mathrm{MX}_{3}\) is prepared according to the following unbalanced chemical equation. $$ \mathrm{M}+\mathrm{X}_{2} \longrightarrow \mathrm{MX}_{3} $$ A \(0.105-g\) sample of \(X_{2}\) contains \(8.92 \times 10^{20}\) molecules. The compound \(\mathrm{MX}_{3}\) consists of \(54.47 \% \mathrm{X}\) by mass. What are the identities of \(\mathrm{M}\) and \(\mathrm{X}\), and what is the correct name for \(\mathrm{MX}_{3}\) ? Starting with \(1.00 \mathrm{~g}\) each of \(\mathrm{M}\) and \(\mathrm{X}_{2}\), what mass of \(\mathrm{MX}_{3}\) can be prepared?

Short answer

The identities of M and X are Strontium (Sr) and Chlorine (Cl), respectively, making the ionic compound SrCl₃, called Strontium Chloride. Starting with 1g each of Sr and Cl₂, 0.919g of SrCl₃ can be prepared.

Step-by-step solution

Determine the molar mass of X2

Since we know the mass and the number of molecules of X2, we can calculate the molecular mass and find out the identity of X. Molecular mass of X2 = (Mass of X2) / (Number of moles of X2) 1 mole = \(6.022 \times 10^{23}\) molecules. So, Number of moles of X2 = (Number of molecules of X2) / (Avogadro's number) = \(8.92 \times 10^{20}\) molecules / \(6.022 \times 10^{23}\) molecules/mole = 1.4818 x 10^(-3) moles of X2 Now, plug the values into the formula: Molecular mass of X2 = 0.105 g / 1.4818 x 10^(-3) moles = 70.8 g/mol Since there are two atoms of X in X2, the molar mass of X = 70.8 g/mol / 2 = 35.4 g/mol. The element with this approximate molar mass is Chlorine (Cl).

Determine the molar mass of M

We know that in compound MX3, the mass percentage of X is 54.47%. Therefore, the mass percentage of M is 100% - 54.47% = 45.53%. Let the molar mass of one mole of MX3 be x g/mol. We can now write the equation using mass percentages and molar masses: (3 x (Molar mass of X) / (Molar mass of MX3)) x 100 = 54.47 (3 x 35.4) / x = 0.5447 Now, solve for x: x = (3 x 35.4) / 0.5447 x = 194.74 g/mol (Molar mass of MX3) To find the molar mass of M, subtract the mass of three moles of Cl: Molar mass of M = 194.74 g/mol - (3 x 35.4 g/mol) = 194.74 g/mol - 106.2 g/mol = 88.54 g/mol The element with this approximate molar mass is Strontium (Sr).

Name of the ionic compound

We have determined the identities of elements M and X as Strontium (Sr) and Chlorine (Cl), respectively. Therefore, the ionic compound MX3 can be written as SrCl3 and is named Strontium Chloride.

Calculating the mass of MX3 prepared from 1g of M and X2

We start with 1g each of M (Strontium) and X2 (Chlorine), so we need to find the moles of M and X2: Moles of Strontium = mass / molar mass = 1/88.54 = 0.0113 moles Moles of Chlorine = mass / molar mass = 1/70.8 = 0.0141 moles Since the molar ratio of Sr:Cl in SrCl3 is 1:3, we need 0.0113 moles of Sr and 0.0339 moles of Cl. But we only have 0.0141 moles of Cl available. As Cl is present in a lesser amount, it is now the limiting reagent. Now, calculate the mass of SrCl3 that can be prepared using the limiting reagent Cl: moles of SrCl3 = moles of Cl / 3 mass of SrCl3 = moles of SrCl3 x molar mass of SrCl3 = (0.0141 / 3) x 194.74 = 0.919 g Thus, starting with 1g of Strontium and 1g of Chlorine, 0.919 g of Strontium Chloride (SrCl3) can be prepared.

Key concepts

Mole Concept

The mole concept is fundamental in chemistry as it bridges the gap between the microscopic world of atoms and molecules and the macroscopic world we can observe. A "mole" is a specific quantity used to express amounts of a chemical substance. One mole is defined as exactly Avogadro's number of molecules or atoms, specifically, \(6.022 \times 10^{23}\) entities. This number is derived from the number of atoms in 12 grams of carbon-12.In the exercise, the number of molecules of \(X_2\) given is \(8.92 \times 10^{20}\). By using the mole concept, these molecules are converted into moles to find out how much of \(X_2\) is present in gram terms. The conversion is done using the formula,\[ \text{Number of moles} = \frac{\text{Number of molecules}}{\text{Avogadro's number}} \]This transformation provides a quantifiable and usable measure for further calculations in stoichiometry and chemical reactions. Understanding the mole concept ensures precision and clarity in chemical equations and reactions.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It uses the coefficients in balanced chemical equations to determine the proportions of reactants and products.In the original exercise, the chemical reaction is shown as \(\mathrm{M} + \mathrm{X}_{2} \rightarrow \mathrm{MX}_{3}\), which after identification of elements becomes balanced to form \( \mathrm{SrCl}_{3} \). Here, stoichiometry helps determine the amounts of M and X required to produce a specific amount of \( \mathrm{MX}_{3} \). To find the number of moles needed for the reaction, one would use:

• The coefficients from the balanced equation to establish molar ratios.
• Convert masses into moles using molar masses.
• Apply the ratios to find out how much of the compound can be theoretically produced.

This process not only ensures that reactions are efficient but also maximizes the usage of available resources.

Limiting Reagent

In a chemical reaction, the limiting reagent is the substance that is completely consumed first and thus determines the maximum amount of product that can be formed. Identifying the limiting reagent is crucial for calculating yields.For this exercise, both strontium and chlorine are started with 1 gram each. Calculations reveal that the moles of chlorine \((0.0141)\) fall short in comparison to that required to fully react with strontium \((0.0339)\) based on the stoichiometric ratios.Therefore, chlorine is the limiting reagent in this context as it restricts the amount of strontium chloride that can form.Understanding which substance limits a reaction allows chemists to:

• Predict the theoretical yield of the product.
• Optimize the amounts of reactants to avoid waste.
• Adjust conditions to potentially increase product yield.

Chemical Naming

Chemical naming follows systematic rules that provide information on the composition and structure of compounds. This ensures consistency and clarity in chemical communication.In simple ionic compounds like \(\mathrm{SrCl}_{3}\), naming involves stating the name of the metal (cation) followed by the non-metal (anion) modified to end in "-ide." Thus, we have "Strontium Chloride".Knowing the identities of \( \mathrm{M} \) and \( \mathrm{X} \) as Strontium (Sr) and Chlorine (Cl), we apply these naming conventions. It reflects:

• "Sr" for Strontium, the metal component.
• "Cl" modified to "-ide," indicating its chloride form.

Chemical naming provides a universal method for identifying compounds, ensuring scientists and learners alike understand the nature of the substances they work with.

Molar Mass Calculation

Molar mass is a critical factor in converting between the mass of a substance and the amount in moles, enabling practical calculations in chemistry.In the problem, the molar mass of \(X\) is derived by first determining the molar mass of \(\mathrm{X}_{2}\) to be approximately \(70.8 \text{ g/mol}\), meaning each \(X\) atom has a molar mass of \(35.4 \text{ g/mol}\). This corresponds to the element chlorine \((Cl)\).The molar mass of \(\mathrm{MX}_{3} (\text{or SrCl}_3)\) is calculated through known values:

• The molar mass of chlorine as determined.
• The given percentage composition, allowing calculation of strontium's molar mass.

This calculation is integral for determining amounts involved in reactions, illustrating the necessity of accurate molar mass values in various stoichiometric calculations.