Question

A certain transition metal ion \(\left(\mathrm{M}^{\mathrm{n}+}\right)\) forms a compound with oxygen \(\left(\mathrm{M}_{\mathrm{x}} \mathrm{O}_{y}\right)\). The molar mass of the compound is \(250.2 \mathrm{~g} / \mathrm{mol}\). If the charge on the transition metal ion is \(+3,\) what is the identity of the transition metal, \(\mathrm{M} ?\) a. \(\mathrm{Th}\) b. Ti c. Hg d. \(\mathrm{Ru}\) e. \(\mathrm{Ag}\)

Short answer

The correct transition metal M in the given compound MₓO_y is \(b.~\text{Ti}\) (Titanium).

Step-by-step solution

Calculate the molar mass of oxygen

We know that the molar mass of oxygen (O) is approximately 16 g/mol. We also know that the molar mass of the compound M_xO_y is 250.2 g/mol. To find the molar mass of the oxygen in the compound, let's use the provided information and denote the number of oxygen atoms as y: Total molar mass of oxygen in the compound = y * molar mass of oxygen = 16y

Calculate the molar mass of the metal

The total molar mass of the compound M_xO_y is given as 250.2 g/mol. To find the molar mass of the metal in the compound, subtract the molar mass of the oxygen: Molar mass of the metal = total molar mass of the compound - total molar mass of oxygen Molar mass of the metal = 250.2 g/mol - 16y

Determine the number of moles of M in the compound

We are given that the charge on the transition metal ion M is +3. Since the charge on an oxygen ion is always -2, the compound must be neutral. Therefore, the total charge of M in the compound should be equal to the total charge of O in the compound. Let x be the number of moles of M in the compound. Then: 3x = 2y

Calculate the molar mass of the metal M

Now we know that 3x = 2y. So, we can write the molar mass of the metal in terms of x: Molar mass of the metal = 250.2 g/mol - 16y Since y = (3/2)x, we can substitute and get: Molar mass of the metal = 250.2 g/mol - 16(3/2)x Molar mass of the metal = 250.2 g/mol - 24x

Compare the calculated molar mass with the given options

Now that we have the molar mass of the metal in terms of x, we can compare it with the molar mass of the given options and find which transition metal M is: a. Th (Thorium) b. Ti (Titanium) c. Hg (Mercury) d. Ru (Ruthenium) e. Ag (Silver) Since the compound is M_xO_y, with a +3 charge on the metal, Option a (Th) and option e (Ag) can be disregarded, as they don't have a common oxidation state of +3. Now, let's compare the remaining options: b. Ti (Molar mass approximately 47.9 g/mol) c. Hg (Molar mass 200.6 g/mol) d. Ru (Molar mass 101.1 g/mol) Since the compound is M_xO_y, the closest of the calculated molar mass values to the molar mass of one of the given options would be the correct transition metal. The difference values for the remaining options are: b. Ti:abs(47.9x - 24x g/mol ) c. Hg:abs(200.6x - 24x g/mol ) d. Ru:abs(101.1x - 24x g/mol ) We must find the nearest integer value, we can find out that For Ti: abs( 47.9x - 250.2) is the most closer to an integer value, and therefore, the correct transition metal M is: b. Ti (Titanium)

Key concepts

Molar Mass Calculation

Understanding molar mass calculation is crucial when analyzing chemical compounds. The molar mass represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). To determine the molar mass of a compound, sum the molar masses of its constituent elements, which are in turn multiplied by their respective number of atoms in the formula.

For example, in the given exercise, the molar mass of the compound M_xO_y is stated to be 250.2 g/mol. With oxygen's molar mass known to be approximately 16 g/mol, the total molar mass of oxygen in the compound could be represented as 16y, where y is the number of oxygen atoms. By subtracting this value from the total molar mass of the compound, the molar mass of the metal, M, is obtained. This step is fundamental as it correlates the molar masses with the stoichiometry of the metal and oxygen atoms in the compound.

Oxidation States

Oxidation states are integral for understanding how elements combine and form compounds. The oxidation state, or oxidation number, indicates the degree of oxidation of an atom in a chemical compound. Conventionally, it is represented by integers, which can be positive, negative, or zero.

Oxygen typically has an oxidation state of -2. In the exercise, the metal ion M has a +3 charge. For the compound to be electrically neutral, the sum of the oxidation states must be zero. Thus, the positive charge provided by the metal ions must balance the negative charge from the oxygen ions. This requirement allows us to derive a simple formula relating the number of metal atoms (x) to the number of oxygen atoms (y): 3x = 2y. By understanding oxidation states and applying the rule of electrical neutrality, one can deduce the stoichiometry of a chemical compound and further solve for its molar mass and formula.

Chemical Compound Formula

The formula of a chemical compound provides key information about the types and amounts of atoms present in the compound. It is a symbolic representation that includes the symbols of the elements and subscript numerals indicating the number of atoms of each element.

In the transition metal compound M_xO_y, 'x' and 'y' represent the number of atoms of metal M and oxygen, respectively. These are determined by understanding the stoichiometry and the charges involved, which was evident in the solution process where we established the relationship between 'x' and 'y' using oxidation states. The correct identification of a chemical compound formula is a result of careful analysis of both molar mass and oxidation states, which leads us to the correct atom count, and eventually, to the identification of the transition metal, M, from the given choices.