Problem 123 - Chemistry and Chemical Reactivity

Question

A compound composed of iron and carbon monoxide, \(\mathrm{Fe}_{x}(\mathrm{CO})_{y},\) is \(30.70 \%\) iron. What is the empirical formula for the compound?

Step-by-Step Solution

Answer

The empirical formula is \(\mathrm{Fe}_{2}(\mathrm{CO})_{9}\).

Step 1: Convert Percentage to Masses

Assume you have 100 grams of the compound. This means you have 30.70 grams of iron (Fe) because the compound is 30.70% iron. The remaining mass composed of carbon monoxide (CO) is 69.30 grams (100 - 30.70).

Step 2: Calculate Moles of Iron

The atomic mass of iron is approximately 55.85 g/mol. Divide the mass of iron by its atomic mass to find the moles of iron:\[\text{Moles of Fe} = \frac{30.70 \text{ grams}}{55.85 \text{ g/mol}} \approx 0.55 \text{ moles}\]

Step 3: Calculate Moles of Carbon Monoxide

The molar mass of carbon monoxide (CO) is approximately 28.01 g/mol (12.01 for carbon and 16.00 for oxygen). Divide the mass of CO by its molar mass to find the moles of carbon monoxide:\[\text{Moles of CO} = \frac{69.30 \text{ grams}}{28.01 \text{ g/mol}} \approx 2.47 \text{ moles}\]

Step 4: Determine the Ratio of Fe to CO

Find the simplest whole number ratio of moles of Fe to moles of CO by dividing both by the smallest number of moles:\[\text{Ratio} = \frac{0.55}{0.55} : \frac{2.47}{0.55} \approx 1 : 4.49\]This ratio simplifies to approximately 1:4.5. Since empirical formulas use whole numbers, we'll multiply by 2 to clear the fraction:\[2 : 9\]

Step 5: Write the Empirical Formula

Based on the simplified ratio, the empirical formula for the compound is \(\mathrm{Fe}_{2}(\mathrm{CO})_{9}\).

Key Concepts

Percent Composition

Percent composition helps us understand the distribution of elements in a compound. It is crucial for determining empirical formulas, especially when given in percentage.
For example, in a compound with a percent composition of 30.70% iron, this percentage tells us how much iron is in a 100 g sample of the compound. This conversion makes it straightforward to translate percentages into actual weights:

• 30.70% iron indicates there are 30.70 grams of iron in a 100 g sample.
• The remaining percentage, or 69.30%, accounts for the rest of the compound, which consists of other elements or groups.

This foundational understanding of percent composition allows us to bridge the gap between qualitative descriptions and quantitative calculations. This step is critical to set up the calculation of moles properly, leading us to determine the empirical formula.

Moles Calculation

The concept of moles is essential in chemistry for converting mass to a number of atoms or molecules. When you know the mass of an element or compound, you can find out how many moles are present using the molar mass.
In the iron and carbon monoxide compound, we calculate the moles of each component by dividing their masses by their respective atomic or molar masses:

• For iron (Fe), the molar mass is 55.85 g/mol:
  \[\text{Moles of Fe} = \frac{30.70 \text{ grams}}{55.85 \text{ g/mol}} \approx 0.55 \text{ moles}\]
• For carbon monoxide (CO), the molar mass is 28.01 g/mol:
  \[\text{Moles of CO} = \frac{69.30 \text{ grams}}{28.01 \text{ g/mol}} \approx 2.47 \text{ moles}\]

This step is crucial because it gives a concrete way to understand the amounts of each substance in terms of molecules or formula units. These mole values are used to find the simplest ratio called the "empirical formula."

Chemical Formula

A chemical formula indicates the elements in a compound and the ratio of atoms. The empirical formula is a simplified version of this, displaying the smallest whole-number ratio of the components.
In this example, we calculated the number of moles for iron and carbon monoxide and used these to find the ratio.
Upon calculating, we found:

• The initial ratio was approximately 1:4.49, indicating nearly five CO units per iron atom.
• By multiplying the entire ratio by 2, we achieved whole numbers: 2:9.

Thus, the empirical formula \(\text{Fe}_2(\text{CO})_9\)illustrates this ratio. The empirical formula doesn't necessarily represent actual quantities of atoms in a molecule, but it reflects the proportional relationship effectively.

Iron Compound

An iron compound involves iron combined with other elements. The study of such compounds involves investigating their composition, which often uses percent composition as a starting point.
Iron often forms complexes with other elements or groups, like carbon monoxide (CO). Compounds such as these are of significant interest in fields such as solid-state chemistry and metallurgy.
The empirical formula of an iron compound derived from combining iron and CO, \(\text{Fe}_2(\text{CO})_9\), indicates how iron and CO are paired in terms of simplest whole-number ratios.
Understanding this enables chemists to make predictions about compound reactivity, properties, and its potential applications.
By approaching compounds from this perspective, comprehensive insights into their scientific and industrial importance are possible.