Problem 38 - Chemistry and Chemical Reactivity

Question

To find the formula of a compound composed of iron and carbon monoxide, \(\mathrm{Fe}_{x}(\mathrm{CO})_{y^{\prime}}\) the compound is burned in pure oxygen to give \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) and \(\mathrm{CO}_{2} .\) If you burn \(1.959 \mathrm{g}\) of \(\mathrm{Fe}_{x}(\mathrm{CO})_{y}\) and obtain \(0.799 \mathrm{g}\) of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) and \(2.200 \mathrm{g}\) of \(\mathrm{CO}_{2},\) what is the empirical formula of \(\mathrm{Fe}_{x}(\mathrm{CO})_{y} ?\)

Step-by-Step Solution

Answer

The empirical formula is \( \mathrm{Fe} \mathrm{(CO)}_5 \).

Step 1: Calculate Moles of Fe in Fe2O3

Determine the moles of Iron (Fe) present in 0.799 g of Fe2O3. The molar mass of Fe2O3 is 159.69 g/mol. Thus, moles of Fe2O3 = \( \frac{0.799 \text{ g}}{159.69 \text{ g/mol}} = 0.005 \text{ mol} \) of Fe2O3. Since there are 2 atoms of Fe in Fe2O3, moles of Fe = \( 2 \times 0.005 = 0.01 \text{ mol} \).

Step 2: Calculate Moles of CO2

Find the moles of CO2 produced. The molar mass of CO2 is 44.01 g/mol. Moles of CO2 = \( \frac{2.200 \text{ g}}{44.01 \text{ g/mol}} = 0.05 \text{ mol} \).

Step 3: Establish the Ratio of Fe to CO

The moles of Fe in the original compound correspond to 0.01 mol, and the moles of CO2 correspond to 0.05 mol of CO, since CO2 is produced from CO during the reaction. The ratio of moles of Fe to moles of CO is thus \( \frac{0.01}{0.05} = 0.2 \). This simplifies to a ratio of 1:5 when considering whole numbers.

Step 4: Determine Empirical Formula

Based on the ratio from Step 3, the empirical formula of the compound can be written as \( \mathrm{Fe} \mathrm{(CO)}_5 \). This is because for every iron atom there are 5 carbon monoxide groups associated, which aligns with our calculated ratio from the combustion products.

Key Concepts

Chemical Reactions

Chemical reactions are processes where reactants are transformed into products through the breaking and forming of chemical bonds. In this exercise, a compound made of iron and carbon monoxide undergoes a chemical reaction when burned in oxygen. This results in the formation of iron(III) oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)) and carbon dioxide (\(\mathrm{CO}_2\)). This type of reaction is known as a combustion reaction, which typically involves an element or compound reacting with oxygen, releasing energy in the form of heat and light.

During the reaction, atoms are neither created nor destroyed. Instead, they are rearranged. The number of atoms of each element present in the reactants must equal the number in the products, in keeping with the law of conservation of mass.

This is why in our example, the atoms of iron initially in \(\mathrm{Fe}_x(\mathrm{CO})_y\) transform into all the \(\mathrm{Fe}_2\mathrm{O}_3\) produced, and the carbon and oxygen atoms form \(\mathrm{CO}_2\). This rearrangement of atoms and conservation of mass are key components in understanding chemical reactions and predicting the products formed.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It allows us to predict how much of a product will form from given quantities of reactants. In the exercise, stoichiometry helps us determine the empirical formula of a compound based on the amounts of \(\mathrm{Fe}_2\mathrm{O}_3\) and \(\mathrm{CO}_2\) produced when the compound is burned.

To perform stoichiometric calculations, we convert the masses of compounds to moles, which allows us to use the coefficients of a balanced chemical equation to calculate the relationships between reactants and products. This exercise involves converting grams to moles for both \(\mathrm{Fe}_2\mathrm{O}_3\) and \(\mathrm{CO}_2\).

This process highlights the concept of molar ratios which help us understand how multiple species in a reaction are related. For the given compounds, it means analyzing the proportion of iron atoms to carbon monoxide groups that formed \(\mathrm{Fe}_2\mathrm{O}_3\) and \(\mathrm{CO}_2\), respectively. This proportion is what ultimately leads to the empirical formula.

Molar Mass Calculation

The molar mass of a compound is the mass of one mole of its entities (atoms, molecules, ions, etc.), and it's crucial in converting between mass and moles in stoichiometry. In this case, the molar mass of \(\mathrm{Fe}_2\mathrm{O}_3\) is 159.69 g/mol, and for \(\mathrm{CO}_2\), it is 44.01 g/mol.

These values are computed by summing the atomic masses of all the atoms in a molecule. For \(\mathrm{Fe}_2\mathrm{O}_3\), it comprises two iron (\(\mathrm{Fe}\)) atoms each with an atomic mass of approx 55.85 u and three oxygen (\(\mathrm{O}\)) atoms each with an atomic mass of approx 16 u.

Knowing the molar mass of a compound allows us to convert its mass in grams to moles by using the formula:

\[ \text{Moles} = \frac{\text{mass in grams}}{\text{molar mass}} \]

This calculation is essential for finding out how many moles of \(\mathrm{Fe}_2\mathrm{O}_3\) or \(\mathrm{CO}_2\) are produced in the combustion reaction, which further helps in determining the empirical formula of the starter compound.

Combustion Analysis

Combustion analysis is a common technique in analytical chemistry to determine the elemental composition of a compound, especially those containing carbon and hydrogen. During combustion, a sample reacts with oxygen and the combustion products, usually \(\mathrm{CO}_2\) and \(\mathrm{H}_2\mathrm{O}\), are analyzed.

In this exercise, combustion analysis helps determine how much of the original compound \(\mathrm{Fe}_x(\mathrm{CO})_y\) was comprised of carbon monoxide, through the measurement of \(\mathrm{CO}_2\) produced. This involves burning the sample completely in the presence of pure oxygen.

The mass of \(\mathrm{CO}_2\) formed is directly associated with the amount of carbon originally present in the compound. Thus, knowing this mass allows us to approximate the stoichiometric coefficients in \(\mathrm{Fe}_x(\mathrm{CO})_y\), such as the ratio of \(x\) to \(y\), by analyzing the combustion products.

It is a crucial process for deriving the empirical formula of chemical compounds containing carbon, hydrogen, and additional elements.