Problem 71 - Chemistry and Chemical Reactivity

Question

Iron carbonyl can be made by the direct reaction of iron metal and carbon monoxide. $$ \mathrm{Fe}(\mathrm{s})+5 \mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{Fe}(\mathrm{CO})_{5}(\ell) $$ What is the theoretical yield of \(\mathrm{Fe}(\mathrm{CO})_{5}\) if \(3.52 \mathrm{g}\) of iron is treated with CO gas having a pressure of \(732 \mathrm{mm} \mathrm{Hg}\) in a \(5.50-\mathrm{L}\) flask at \(23^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The theoretical yield of \(Fe(CO)_5\) is 8.58 g.

Step 1: Calculate Moles of Iron

First, find the molar mass of iron (Fe), which is approximately 55.85 g/mol. Then, calculate the number of moles of iron used:\[moles\,of\,Fe = \frac{3.52\,g}{55.85\,g/mol} \approx 0.0630 \,mol\]

Step 2: Use Ideal Gas Law to Calculate Moles of CO

The ideal gas law, \(PV = nRT\), will help calculate moles of CO gas. Convert pressure to atm by dividing by 760 mm Hg/atm:\[P = \frac{732\,mm\,Hg}{760\,mm\,Hg/atm} \approx 0.963\,atm\]Convert temperature to Kelvin:\[T = 23^{\circ}C + 273.15 = 296.15 \, K\]Using \(R = 0.0821\,L\,atm/mol\,K\) in the ideal gas law, solve for moles of CO:\[0.963 \, atm \times 5.50 \, L = n \times 0.0821 \, L \, atm/mol \times 296.15 \,K\]\[ n = \frac{0.963 \times 5.50}{0.0821 \times 296.15} \approx 0.219\,mol\]

Step 3: Determine Limiting Reactant

From the balanced equation, \(1\,mol\,Fe\) reacts with \(5\,mol\,CO\). Compare the mole ratio:- Fe: 0.0630 mol- Required CO for 0.0630 mol Fe = \(0.0630 \,mol \times 5 = 0.315 \,mol\).Since only 0.219 mol CO is available, CO is the limiting reactant.

Step 4: Calculate Theoretical Yield of Iron Carbonyl

One mole of CO produces \(\frac{1}{5}\) mole of \(Fe(CO)_5\):Using the moles of CO (limiting reactant):\[moles\,of\,Fe(CO)_5 = \frac{0.219\,mol\,CO}{5} = 0.0438 \,mol\]Calculate the molar mass of \(Fe(CO)_5\), approximately 195.9 g/mol:Theoretical yield in grams:\[0.0438\,mol \times 195.9\,g/mol \approx 8.58\,g\]

Step 5: Conclusion

The theoretical yield of iron carbonyl, \(Fe(CO)_5\), is calculated based on the limiting reactant, CO. Given the stoichiometry and moles of reactants, the yield is limited by CO available.

Key Concepts

Stoichiometry

Stoichiometry is the quantitative analysis of reactants and products in a chemical reaction. It involves using the balanced chemical equation to determine the relationship between the amounts of substances consumed and produced.
In the context of this exercise, stoichiometry helps us determine how much iron carbonyl (\(Fe(CO)_5\)) we can theoretically obtain by reacting iron (\(Fe\)) with carbon monoxide (\(CO\)).

• Begin with a balanced equation: \( \text{Fe(s)} + 5\, \text{CO(g)} \rightarrow \text{Fe(CO)}_5(l) \)
• The coefficients in the equation indicate that 1 mole of iron reacts with 5 moles of carbon monoxide to produce 1 mole of iron carbonyl.
• By finding out how many moles of each reactant are present, we can predict the number of moles of \(Fe(CO)_5\) formed.

A key part of stoichiometry is ensuring all units are consistent, and conversions between grams, moles, and liters are frequently required. It's important to use molar masses and the ideal gas law as necessary to make these conversions.

Limiting Reactant

A limiting reactant is the substance that is completely used up first in a chemical reaction, thus dictating the amount of product that can be formed. This is crucial because it limits the extent to which a reaction can proceed.
In this exercise, we determined that carbon monoxide is the limiting reactant. Here's how we can identify the limiting reactant:

• Calculate moles of each reactant present. For iron, we calculated approximately 0.0630 moles, while for carbon monoxide, we found about 0.219 moles.
• Using the moles obtained and the balanced equation, determine the mole ratio and required amount. We know from stoichiometry that we need 5 moles of CO for every mole of Fe.
• Compare the required moles to the available moles: We need 0.315 moles of CO for the 0.0630 moles of Fe, but we only have 0.219 moles of CO.
• Since CO is less than the required amount, CO is the limiting reactant, dictating the theoretical yield of \(Fe(CO)_5\).

Understanding which reactant limits the reaction is essential for calculating theoretical yields.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and amount of gas using the formula \( PV = nRT \). This equation is instrumental in determining the number of moles of a gas when some of these variables are known.
For this exercise:

• Convert the given pressure to atmospheres (atm) by dividing by the constant 760 mm Hg/atm: \(\frac{732 \, mm \, Hg}{760} \approx 0.963 \, atm\).
• Convert the Celsius temperature to Kelvin by adding 273.15: \(23^{\circ}C + 273.15 = 296.15 \, K\).
• Use the equation to solve for the number of moles \(n\): \[0.963 \, atm \times 5.50 \, L = n \times 0.0821 \, L \, atm/mol \times 296.15 \, K\]
• Solving the equation gives \(n \approx 0.219 \text{ moles of CO}.\)

The ideal gas law is invaluable for calculating quantities of gases in reactions, even under non-standard conditions.

Molar Mass

Molar mass is the mass of one mole of a given substance, measured in grams per mole (g/mol), and is used to convert between mass and moles, a crucial step in stoichiometry.
For this problem, knowing the molar masses of the substances involved is fundamental:

• Iron (Fe) has a molar mass of approximately 55.85 g/mol.
• Carbon monoxide (CO) is 28.01 g/mol, though not directly used in our calculation.
• To calculate the theoretical yield, we use the molar mass of iron carbonyl \(Fe(CO)_5\), which is approximately 195.9 g/mol.

Once we determined the limiting reactant, we used the molar mass of \(Fe(CO)_5\) to find the theoretical yield. By multiplying the moles of \(Fe(CO)_5\) (0.0438 mol) by its molar mass (195.9 g/mol), we found the theoretical yield: 8.58 g. Understanding and using molar mass is key to moving between quantities in chemical equations.