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 \(23^{\circ} \mathrm{C} ?\)

Short answer

The theoretical yield is 8.54 g of Fe(CO)_5.

Step-by-step solution

Calculate Moles of Iron

First, calculate the number of moles of iron. The molar mass of iron (Fe) is \(55.85\ \text{g/mol}\). Use the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\).\[moles\ \text{ of } Fe = \frac{3.52\ \text{g}}{55.85\ \text{g/mol}} = 0.063\ \text{mol}\]So, the moles of iron is \(0.063\ \text{mol}\).

Calculate Moles of CO

Use the ideal gas law to calculate the moles of CO gas. The ideal gas law is \(PV = nRT\), where \(P\) is pressure in atm, \(V\) is volume in L, \(n\) is moles, \(R\) is the gas constant \(0.0821\ \text{L atm/mol K}\), and \(T\) is the temperature in Kelvin.First, convert pressure from mm Hg to atm:\[P = \frac{732\ \text{mm Hg}}{760\ \text{mm Hg/atm}} = 0.963\ \text{atm}\]Next, convert temperature to Kelvin:\[T = 23 + 273.15 = 296.15\ \text{K}\]Now, solve for \(n\):\[0.963\ \text{atm} \times 5.50\ \text{L} = n \times 0.0821\ \text{L atm/mol K} \times 296.15\ \text{K}\]\[n = \frac{0.963 \times 5.50}{0.0821 \times 296.15} \approx 0.218\ \text{mol CO}\]

Determine Limiting Reactant

In the reaction, 1 mole of Fe requires 5 moles of CO. Compare the available mole ratio:For Fe:\(0.063\ \text{mol Fe} \rightarrow 0.315\ \text{mol CO needed (0.063 mol} \times 5)\)For CO:\(0.218\ \text{mol CO available}\)Since 0.315 mol of CO is needed, but only 0.218 mol is available, CO is the limiting reactant.

Calculate Theoretical Yield of Fe(CO)_5

According to the reaction, 5 moles of CO yield 1 mole of \(\text{Fe(CO)}_5\). As CO is the limiting reactant, use its moles to find the yield:\[\text{Moles of } \text{Fe(CO)}_5 = \frac{0.218\ \text{mol CO}}{5} = 0.0436\ \text{mol Fe(CO)}_5\]The molar mass of \(\text{Fe(CO)}_5\) is \(195.9\ \text{g/mol}\), so:\[\text{Mass of } \text{Fe(CO)}_5 = 0.0436\ \text{mol} \times 195.9\ \text{g/mol} = 8.54\ \text{g}\]Therefore, the theoretical yield of \(\text{Fe(CO)}_5\) is \(8.54\ \text{g}\).

Key concepts

Limiting Reactant

When it comes to chemical reactions, the concept of the limiting reactant is crucial. This is because it determines the maximum amount of product that can be formed. The limiting reactant is the substance that is completely consumed when the reaction goes to completion. In our example, we have iron (Fe) and carbon monoxide (CO) reacting to form iron carbonyl. To find out which reactant is limiting, we must first calculate the amount of each reactant available and see which one runs out first. This involves comparing the mole ratios. For our reaction, 1 mole of iron requires 5 moles of carbon monoxide to complete the reaction as per the equation: Fe + 5 CO → Fe(CO)₅. In the given problem, the moles of iron are 0.063, which would require 0.315 moles of CO. However, we only have 0.218 moles of CO available, making carbon monoxide the limiting reactant. Since there is not enough CO to convert all of the available iron into product, CO limits the reaction and thus determines the theoretical yield of iron carbonyl.

Ideal Gas Law

The ideal gas law is an essential tool for connecting the physical properties of gases: pressure (P), volume (V), and temperature (T) with the amount of gas in moles (n). It is expressed as:\(PV = nRT\).In this equation, \(R\) is the ideal gas constant, which equals 0.0821 L atm/mol K. To solve for the number of moles in a gas sample, rearrange the equation to: \(n = \frac{PV}{RT}\).Let's apply this to find the moles of CO in our reaction. We start by converting the pressure from mm Hg to atm, as calculations using the ideal gas law require standard units.1 atm = 760 mm Hg, so \(732 \text{ mm Hg} = 0.963 \text{ atm}\).Also, the temperature needs to be in Kelvin:\(T(\text{K}) = 23 + 273.15 = 296.15 \text{ K}\).Now we can find the moles of CO:\(n = \frac{(0.963 \text{ atm})(5.50 \text{ L})}{(0.0821 \text{ L atm/mol K})(296.15 \text{ K})} \approx 0.218 \text{ mole CO}\).This calculation correctly shows that the amount of CO available for the reaction is 0.218 moles, another hint towards identifying the limiting reactant.

Molar Mass Calculation

To understand productivity in chemical reactions, knowing how to calculate molar mass is essential. Molar mass, often measured in grams per mole, is important for converting between the mass of a substance and its moles.To calculate it:- Add up the atomic masses of all elements in the compound (using the periodic table for atomic masses).For instance, iron carbonyl \(\text{Fe(CO)}_5\) requires knowledge of atomic masses of iron and carbon monoxide. The molar mass of iron is 55.85 g/mol, and carbon (12.01 g/mol) plus oxygen (16.00 g/mol) gives CO a molar mass of 28.01 g/mol. Therefore, for \(\text{Fe(CO)}_5\):- Calculate as: \(55.85 \text{ g/mol for Fe} + 5 \times 28.01 \text{ g/mol for CO} = 195.9 \text{ g/mol}\).This molar mass helps determine both the moles of \(\text{Fe(CO)}_5\) possible from limiting CO and ultimately compute the theoretical yield (i.e., predicted mass of product resulting from a complete reaction of limiting reactant). In our example case, with 0.0436 moles of \(\text{Fe(CO)}_5\) produced and a molar mass of 195.9 g/mol, the theoretical yield is 8.54 grams.