Problem 100 - General Chemistry: Principles and Modern Applications

Question

A 0.622 g sample of a metal oxide with the formula \(\mathrm{M}_{2} \mathrm{O}_{3}\) is converted to \(0.685 \mathrm{g}\) of the sulfide, MS. What is the atomic mass of the metal M?

Step-by-Step Solution

Answer

To get the short answer, solve the final equation obtained in Step 3. That will give the atomic mass of the metal M in grams per mole.

Step 1: Derive the expression for mass in metal oxide

The formula for metal oxide is provided as M2O3. This means that in one mole of metal oxide, there are 2 moles of the metal M. Given that the molar mass is equal to grams per mole, the atomic mass of M in the metal oxide can be represented as follows: \(2M_{mass} + 3(16) = 0.622g\) (where 16 is the atomic mass of oxygen).

Step 2: Derive the expression for mass in metal sulfide

The formula for metal sulfide is MS, meaning that in one mole of metal sulfide, there's one mole of the metal M. The atomic mass of M in the metal sulfide can therefore be represented as follows: \(M_{mass} + 32.07 = 0.685g\) (where 32.07 is the atomic mass of sulfur).

Step 3: Solve the system of equations

Now, there are two equations with M_mass as the variable. This system of equations can be solved by substitution or elimination method. Let’s choose substitution in this case, isolate M_mass in the second equation and substitute it into the first equation to find the atomic mass of M.

Key Concepts

Understanding Metal Oxides

Metal oxides are compounds composed of metal cations bonded with oxygen anions. They are typically formed by the reaction of metals with oxygen and can exist as solids at room temperature.
In chemistry, metal oxides often have the general formula \((M_xO_y)\), where \(M\) represents the metal, and \(O\) represents oxygen.
Key characteristics to keep in mind:

• Metal oxides usually exhibit ionic bonding due to the transfer of electrons from the metal to the oxygen.
• These compounds are important in various industries, including manufacturing and materials science.
• The formula of a metal oxide indicates the stoichiometry, or ratio, of metal to oxygen atoms in the compound.

The formula \(\mathrm{M}_{2} \mathrm{O}_{3}\) in our exercise tells us that there are two metal atoms for every three oxygen atoms. Understanding how to interpret these formulas is crucial for solving problems related to metal oxides.

Deciphering Chemical Formulas

Chemical formulas communicate the types and numbers of atoms in a compound. They serve as a shorthand notation for representing molecular and ionic structures.
For example, the chemical formula \(\mathrm{M}_{2}\mathrm{O}_{3}\) signifies that in a single molecule or formula unit of the metal oxide, there are two metal atoms and three oxygen atoms.
Consideration points:

• The subscripts in a chemical formula provide the ratio of each type of atom in the compound. No subscript next to an element symbol means there is only one atom of that element in the compound.
• Chemical formulas can be derived from empirical or molecular formulas, with the former showing the simplest integer ratio of atoms.
• In ionic compounds like metal oxides, the overall charge of the formula must balance to zero, ensuring charge neutrality.

Converting from one chemical compound to another, such as from an oxide \(\mathrm{M}_{2}\mathrm{O}_{3}\) to a sulfide \(MS\), shows how chemical formulas help in tracking stoichiometry processes during reactions.

Breaking Down Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It allows chemists to predict how much of each substance is required or produced in a given reaction.
The foundational concept of stoichiometry is the mole, which is a measure that allows chemists to count atoms, molecules, or other entities using Avogadro's number (approximately \(6.022 \times 10^{23}\) entities per mole).
In our exercise:

• The stoichiometry of the reaction is key to relating the mass of the metal oxide (0.622 g) to the mass of metal sulfide (0.685 g).
• Using the formulas \(\mathrm{M}_{2}\mathrm{O}_{3}\) and \(MS\), we can set up equations to solve for the atomic mass of the metal \(M\).
• By writing chemical equations and using molar masses of elements like oxygen and sulfur, we balance and solve for unknowns.

Stoichiometry acts as a bridge between the reactants and products, helping in calculating the needed quantities for balanced chemical reactions, ensuring that the conversion from reactant to product is accurately represented.