Question

A 0.370 -mol sample of a metal oxide \(\left(\mathrm{M}_{2} \mathrm{O}_{3}\right)\) weighs \(55.4 \mathrm{~g}\). (a) How many moles of \(\mathrm{O}\) are in the sample? (b) How many grams of \(\mathrm{M}\) are in the sample? (c) What element is represented by the symbol M?

Short answer

(a) 1.11 moles of \(\text{O}\). (b) 37.64 g of \(\text{M}\). (c) The element is vanadium.

Step-by-step solution

Determine the number of moles of \(\text{O}\)

In \(\text{M}_{2}\text{O}_{3}\), every molecule contains 3 atoms of oxygen. Therefore, the number of moles of oxygen is three times the number of moles of \(\text{M}_{2}\text{O}_{3}\). Since we have a 0.370-mol sample of \(\text{M}_{2}\text{O}_{3}\), the moles of oxygen \(O\) in the sample is \[ 3 \times 0.370 \text{ moles } = 1.11 \text{ moles } \]

Calculate molar mass of \(\text{M}_{2}\text{O}_{3}\)

Given that the sample weighs 55.4 g and consists of 0.370 moles, use the molar mass formula \[ \text{Molar mass} = \frac{\text{mass}}{\text{moles}} \] to find the molar mass of \(\text{M}_{2}\text{O}_{3}\) : \[ \text{Molar mass of M}_2\text{O}_3 = \frac{55.4 \text{ g}}{0.370 \text{ mol}} = 149.73 \text{ g/mol} \]

Separate molar masses of \(\text{M}\) and \(\text{O}\)

The molar mass of \(\text{M}_{2}\text{O}_{3}\) is the sum of the molar masses of 2 atoms of \(\text{M}\) and 3 atoms of \(\text{O}\). The molar mass of oxygen is 16.00 g/mol, so the molar mass contribution from oxygen is \[ 3 \times 16.00 \text{ g/mol} = 48.00 \text{ g/mol} \]

Find molar mass of \(\text{M}\)

Subtract the molar mass contribution of oxygen from the total molar mass to find the contribution from \(\text{M}\): \[ 149.73 \text{ g/mol} - 48.00 \text{ g/mol} = 101.73 \text{ g/mol} \] Since this is the total mass for 2 atoms of \(\text{M}\), find the molar mass of one \(\text{M}\) atom by dividing by 2: \[ \text{Molar mass of M} = \frac{101.73 \text{ g/mol}}{2} = 50.87 \text{ g/mol} \]

Identify the element \(\text{M}\)

Using the periodic table, find the element with a molar mass of approximately 50.87 g/mol. This corresponds to vanadium (V), which has an atomic mass of approximately 50.94 g/mol.

Calculate grams of \(\text{M}\) in the sample

To find the mass of \(\text{M}\) in the sample, multiply the number of moles of \(\text{M}_{2}\text{O}_{3}\) by the total molar mass contribution of \(\text{M}\): \[ 0.370 \text{ moles} \times 101.73 \text{ g/mol} = 37.64 \text{ g} \]

Key concepts

Molar Mass Calculation

Molar mass is a fundamental concept in stoichiometry. It is the mass of one mole of a substance, measured in grams per mole (g/mol). To determine the molar mass of a compound, you need both the mass and the number of moles of the sample.

Using the formula for molar mass: \[ \text{Molar mass} = \frac{\text{mass}}{\text{moles}} \] We can plug in the given values from the problem. For instance, given a 0.370-mol sample of \(\text{M}_2\text{O}_3\) with a mass of 55.4 g, we calculate:

\[ \text{Molar mass of M}_2\text{O}_3 = \frac{55.4 \text{ g}}{0.370 \text{ mol}} = 149.73 \text{ g/mol} \]

This result tells us the molar mass of the compound \(\text{M}_2\text{O}_3\) is 149.73 g/mol.

Chemical Formula Interpretation

Understanding chemical formulas is crucial for solving stoichiometry problems. A chemical formula provides the ratio of each type of atom in a compound. For example, \(\text{M}_2\text{O}_3\) shows that for every 2 atoms of element M, there are 3 atoms of oxygen (O).

By analyzing the chemical formula, we can determine the number of moles of each element present in the compound. In our problem, the sample has 0.370 moles of \(\text{M}_2\text{O}_3\). Since each molecule contains 3 oxygen atoms, the moles of oxygen in the sample are:

\[ 3 \times 0.370 \text{ moles} = 1.11 \text{ moles} \]

This means there are 1.11 moles of oxygen in our sample of 0.370 moles of \(\text{M}_2\text{O}_3\).

Element Identification

Identifying an element from its properties, such as molar mass, involves comparing calculated values with known values from the periodic table. For our compound \(\text{M}_2\text{O}_3\), we first determine the molar mass of each element. The molar mass of oxygen (O) is 16.00 g/mol. With 3 oxygen atoms, the contribution to the total molar mass is:

\[ 3 \times 16.00 \text{ g/mol} = 48.00 \text{ g/mol} \]

Subtracting the oxygen's contribution from the total molar mass of \(\text{M}_2\text{O}_3\), we get the molar mass of M:

\[ 149.73 \text{ g/mol} - 48.00 \text{ g/mol} = 101.73 \text{ g/mol} \]

Since this is for 2 atoms of M, the molar mass of one atom is:

\[ \frac{101.73 \text{ g/mol}}{2} = 50.87 \text{ g/mol} \]

By inspecting a periodic table, we find that the element with a molar mass close to 50.87 g/mol is vanadium (V), which has a molar mass of approximately 50.94 g/mol. Thus, element M is vanadium.

To calculate the grams of vanadium in the sample, we multiply the moles of \(\text{M}_2\text{O}_3\) by the molar mass contribution from V:

\[ 0.370 \text{ moles} \times 101.73 \text{ g/mol} = 37.64 \text{ g} \]

Therefore, the sample contains 37.64 grams of vanadium (V).