Question

Consider an equimolar mixture (equal number of moles) of two diatomic gases \(\left(\mathrm{A}_{2}\right.\) and \(\mathrm{B}_{2}\) ) in a container fitted with a piston. The gases react to form one product (which is also a gas) with the formula \(\mathrm{A}_{x} \mathrm{~B}_{y}\). The density of the sample after the reaction is complete (and the temperature returns to its original state) is \(1.50\) times greater than the density of the reactant mixture. a. Specify the formula of the product, and explain if more than one answer is possible based on the given data. b. Can you determine the molecular formula of the product with the information given or only the empirical formula?

Short answer

We cannot specify a unique empirical formula of the product based on the given data as there are multiple possible combinations that fit the criteria. With the given information, we can only partially determine the empirical formula of the product, but not the molecular formula.

Step-by-step solution

Calculate the molar mass of the reactant mixture

Since the gases are diatomic, their molar mass would be 2A and 2B (where A and B represent the molar masses of elements A and B). The reactant mixture is equimolar, which means there are equal amounts of A₂ and B₂. Therefore, the average molar mass of the reactant mixture is: Average molar mass = \(\frac{1}{2}(2A + 2B)\)

Calculate the molar mass of the product

The question states that the density of the sample after the reaction is 1.50 times greater than the density of the reactant mixture. Since density is related to molar mass: Molar mass of the product = 1.50 × Average molar mass of the reactant mixture Molar mass of the product = 1.50 × \(\frac{1}{2}(2A + 2B)\) Molar mass of the product = \(1.50(A + B)\)

Find the empirical formula

The empirical formula of the product is AxBy. We know that the molar mass of the product is 1.50(A+B). Since the only information given is regarding the molar mass ratio of the product, we cannot determine specific values for x and y. However, we can check if more than one answer is possible based on the given data.

Determine if more than one answer is possible

There is more than one possibility for the formula of the product based on the given data. For example, it could be: A₁B₁ (one atom of A and one atom of B in the product) A₂B₂ (two atoms of A and two atoms of B in the product) A₃B₃ (three atoms of A and three atoms of B in the product) All these combinations could result in the molar mass of the product being 1.50 times greater than the molar mass of the reactant mixture. Result: We cannot specify the unique empirical formula of the product based on the given data as there are multiple possible combinations that fit the criteria. #b. Molecular Formula or Empirical Formula#

Analyze the given data

We only have information about the molar mass ratio between the product and the reactant mixture. We do not have any information about the actual elements A and B or their reaction stoichiometry.

Determine if the molecular formula can be deduced

Given the limited information, we cannot determine the molecular formula of the product, as we need additional data about the elements (A and B) and their stoichiometry in the reaction. Result: With the given information, we can only partially determine the empirical formula of the product, but not the molecular formula.

Key concepts

Molar Mass

The molar mass of a substance is a crucial concept in understanding gas reactions. It refers to the mass of one mole of a chemical substance. For diatomic gases like \(\mathrm{A}_2\) and \(\mathrm{B}_2\), the molar mass is calculated as twice the atomic mass of the individual elements since each molecule contains two atoms. In the context of our exercise, both gases \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are considered to contribute equally to the mixture since it's equimolar.

When calculating the average molar mass of the reactant mixture, we consider both components equally due to the equimolar nature. This gives us the equation: \[\frac{1}{2}(2A + 2B)\]

Understanding the molar mass is foundational because it affects other properties such as the density of gases, which is directly relevant in this problem since the reaction's product has a denser composition than the sum of its reactants.

Density

Density is a measure of how much mass is contained in a given volume. In gas reactions, density offers insights into the characteristics of the products formed. When the problem states that the product's density is 1.50 times that of the reactant mixture, it is indicating that the product has packed more mass into the same volume.

The density of a gas is related to its molar mass and pressure, among other variables. A higher molar mass at constant temperature and pressure will generally contribute to an increase in density.

In this exercise, understanding density informs us not just about the relative mass but links directly back to determining the potential formulas of the gas formed. The increase by 1.50 times helps approximate the molar mass of the new gas product, moving from reactants to products.

Empirical Formula

The empirical formula is a simple statement of the ratio of each type of atom within a compound. It's the basic formula that reflects the minimum number of atoms in their simplest whole-number ratio.

For this gas reaction, the task is to determine the empirical formula given that the ratio of molar masses changes to 1.50. While we know the molar mass of the resultant product in terms of \(A + B\), the precise empirical formula is elusive without specific measurements of the individual elements \(A\) and \(B\).

Empirically, the formula will satisfy this multiplicative relationship, as given by 1.5 times the reactant mass. The formulas \(A_1B_1\), \(A_2B_2\), or even \(A_3B_3\) serve as possible candidates because they all fit the general formula's possible combinations based on the current data.

Molecular Formula

The molecular formula tells us how many atoms of each element are present in a molecule of a compound. Compared to the empirical formula, the molecular formula provides the real, whole number ratio of atoms of each element in a compound, not just the simplest form.

In this problem, while we can calculate the proportional increase in molar mass, what we lack is direct information on \(A\) and \(B\) themselves, or their exact combining ratios. Thus, determining the molecular formula involves additional key information such as structural components and accurate elemental counts which are unavailable here.

Therefore, in our case, with limited data focusing just on comparative density and molar mass, the systematic determination of a unique molecular formula remains unresolved. It is, however, a possibility if more detailed elemental or stoichiometric data could be provided.