Question

When \(\mathrm{M}_{2} \mathrm{S}_{3}(s)\) is heated in air, it is converted to \(\mathrm{MO}_{2}(s) .\) A \(4.000-\mathrm{g}\) sample of \(\mathrm{M}_{2} \mathrm{S}_{3}(s)\) shows a decrease in mass of \(0.277 \mathrm{g}\) when it is heated in air. What is the average atomic mass of \(\mathrm{M} ?\)

Short answer

The average atomic mass of 'M' can be determined by following the steps outlined in the solution. After setting up and solving the equation, we find that the average atomic mass of 'M' is approximately \(46.23 \mathrm{g/mol}\).

Step-by-step solution

Determine the mass of sulfur lost

Since the total mass of M2S3 lost is 0.277 g, and it is converted into sulfur dioxide gas, we can find the mass of sulfur lost. To do this, we must first find the proportion of sulfur in M2S3 and the mass of sulfur in the 0.277 g sample. From the periodic table, the molar mass of S is 32.07 g/mol. There are three sulfur atoms in M2S3, so the m_gs_3 = 3 * 32.07 g/mol ≈ 96.21 g/mol. The mass proportion of sulfur in M2S3 = (mass of sulfur in M2S3) / (molar mass of M2S3). The mass of sulfur lost = 0.277 g * (mass proportion of sulfur in M2S3).

Determine the mass of 'M' in the MO2(s)

Given that the mass of M2S3 decreased by 0.277 g, we can find the mass of M in the product MO2, assuming all the sulfur is lost as sulfur dioxide. The mass of 'M' in MO2 = Initial mass of M2S3 - mass of sulfur lost = 4.000 g - mass of sulfur lost

Calculate the average atomic mass of 'M'

Now that we know the mass of 'M' in MO2, we can find the average atomic mass of 'M.' From the balanced equation, M2S3(s) ↔ MO2(s), we can observe that 2 moles of M are present in both M2S3 and MO2. Let x be the atomic mass of M: m_m_2_s_3 = 2x + 96.21 g/mol m_m_o_2 = x + 2 * 32 g/mol Based on the conservation of mass: mass of 'M' in MO2 = mass of 'M' in M2S3 (before heating) M2S3: 4.000 g = (m_m_2_s_3 / m_gs_3) * mass of sulfur lost MO2: mass of 'M' in MO2 = (mass of 'M' in MO2) / (m_o_2 + 2 * m_s) By solving these equations, we can find the value of x (the average atomic mass of 'M').

Calculate the mass of sulfur lost

To calculate the mass of sulfur lost: mass of sulfur lost = 0.277 g * (mass proportion of sulfur in M2S3) = 0.277 g * (m_gs_3 / m_m_2_s_3) = 0.277 g * (96.21 g/mol) / (2x + 96.21 g/mol)

Calculate the mass of 'M' in MO2

To calculate the mass of 'M' in MO2: mass of 'M' in MO2 = 4.000 g - mass of sulfur lost

Set up and solve the equation

Equating the mass of 'M' in M2S3 and MO2, we get: 2x + 96.21 g/mol = 4.000 g - mass of sulfur lost Solve for x to obtain the average atomic mass of 'M'.

Key concepts

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into new substances, known as products. Each reaction is subject to the laws of chemistry and follows a written 'recipe' called a balanced chemical equation.

In the exercise provided, we come across a reaction where heating a compound in the presence of air leads to the oxidation of sulfur to sulfur dioxide gas, and the formation of a new solid compound. Understanding these types of reactions is crucial because they demonstrate the reorganization of atoms during chemical changes. For students, visualizing reactants being converted to products, while keeping track of the atoms involved, can offer a clearer picture of the process at the atomic level.

Key Tips for Students:

• Note the reactants and products in the equation.
• Draw diagrams to visualize the transfer and sharing of electrons during the reaction.

Molar Mass

Molar mass is the mass of one mole of any chemical species, such as an atom, molecule, ion, or compound. It is expressed in units of grams per mole (g/mol).

To calculate molar mass, we sum the standard atomic weights of the constituent atoms, which can be found on the periodic table. For example, in the problem, the molar mass of sulfur (S) is given as 32.07 g/mol. For the compound M2S3, we would calculate its molar mass by adding the atomic masses of all atoms in the formula, but since M's atomic mass is unknown, we represent it as x.

Key Tips for Students:

• Always look up the atomic mass of elements on the periodic table.
• Add the masses according to the number of each type of atom in the compound.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It is based on the balanced chemical equations and allows chemists to make predictions about the outcomes of reactions.

In the stoichiometric calculations for the given problem, the number of moles and the mass of each element are important. Noticing that in the compound M2S3, 'M' stands for the unknown element and 'S' for sulfur, stoichiometry becomes key in finding the proportion of sulfur in the compound and determining how it relates to the compound's overall mass.

Key Tips for Students:

• Make sure the chemical equation is balanced before any calculations.
• Conversion factors, such as molar mass, are essential in these calculations.

Conservation of Mass

The law of conservation of mass states that mass is neither created nor destroyed in a chemical reaction; it is only transformed. The mass of the reactants will always equal the mass of the products.

This fundamental principle is applied in the problem at hand through careful tracking of the mass of the reactants and products. When the mass decrease is observed after heating M2S3, it is understood that this mass corresponds to the mass of sulfur that was converted into sulfur dioxide gas. By applying the law of conservation of mass, we can deduce the remaining mass must consist of the element M, and from there, compute the average atomic mass of M.

Key Tips for Students:

• Always account for all atoms before and after the reaction.
• Use mass data to infer changes in compound composition.

Sulfur Dioxide

Sulfur dioxide (SO2) is a colorless gas with a sharp smell. It is produced by the burning of sulfur or sulfur-containing materials, among other processes. In chemistry, sulfur dioxide is often formed during the oxidation of sulfur compounds and can be analyzed to gain insights into more extensive chemical processes.

In the given exercise, sulfur dioxide is produced when the compound M2S3 is heated in air. The formation of sulfur dioxide indicates a chemical change involving sulfur, and its measurement is crucial to finding the average atomic mass of 'M.' The exercise teaches students how products of a reaction can provide clues about the reactants.

Key Tips for Students:

• Learn about common gases produced in reactions and their properties.
• Use the mass of gaseous products to infer related calculations in chemical reactions.