Question

\(\mathrm{Ni}(\mathrm{CO})_{4}\) can be made by reacting finely divided nickel with gaseous CO. If you have CO in a \(1.50-\mathrm{L}\). flask at a pressure of \(418 \mathrm{mm}\) Hg at \(25.0^{\circ} \mathrm{C},\) along with \(0.450 \mathrm{g}\) of Ni powder, what is the theoretical yield of \(\mathrm{Ni}(\mathrm{CO})_{4} ?\)

Short answer

The theoretical yield of \( \mathrm{Ni}(\mathrm{CO})_4 \) is 1.31 g.

Step-by-step solution

Write the Balanced Equation

The balanced chemical equation for the reaction between Nickel (Ni) and Carbon Monoxide (CO) to form Nickel Carbonyl \(\mathrm{Ni}(\mathrm{CO})_4\) is as follows: \\[ \mathrm{Ni}_{(s)} + 4\mathrm{CO}_{(g)} \rightarrow \mathrm{Ni}(\mathrm{CO})_4 \] \This indicates that one mole of Nickel reacts with four moles of Carbon Monoxide to produce one mole of \(\mathrm{Ni}(\mathrm{CO})_4\).

Calculate Moles of Reactants

We know that number of moles \(n\) is given by \( n = \frac{m}{M} \) where \(m\) is mass and \(M\) is molar mass. \Molar mass of Ni is \(58.69\, \text{g/mol}\). Therefore, moles of Ni = \( \frac{0.450\, \text{g}}{58.69\, \text{g/mol}} = 0.00767\, \text{mol}\). \Using the Ideal Gas Law \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(R\) is the ideal gas constant \(0.0821\, \text{L atm/mol K}\), and \(T\) is temperature in Kelvins. First, convert pressure from mmHg to atm, where \(418\, \text{mm Hg} = \frac{418}{760} \approx 0.550\, \text{atm}\) and \(T = 25.0 + 273.15 = 298.15\, \text{K}\). Thus moles of CO \(n\) are calculated as: \\[ n = \frac{PV}{RT} = \frac{(0.550)(1.50)}{(0.0821)(298.15)} = 0.0336\, \text{mol} \]

Determine the Limiting Reactant

According to the balanced equation, 4 moles of CO are required for every mole of Ni. We have \(0.00767\, \text{mol}\, \text{Ni}\) requiring \(4 \times 0.00767 = 0.0307\, \text{mol}\, \text{CO}\). \(0.0336\, \text{mol}\, \text{CO}\) is available which is more than \(0.0307\, \text{mol}\), so Ni is the limiting reactant.

Calculate Theoretical Yield of \( \mathrm{Ni}(\mathrm{CO})_4 \)

The limiting reactant \( \text{Ni} \) drives the reaction, and since \( \text{1 mol Ni} \rightarrow \text{1 mol Ni}(\text{CO})_4 \), \( 0.00767\, \text{mol} \, \text{Ni}\) will produce \( 0.00767 \text{ mol Ni}(\text{CO})_4 \). \The molar mass of \( \mathrm{Ni}(\mathrm{CO})_4 \) is \(58.69 + 4(12.01 + 16.00) = 171.0\, \text{g/mol}\). \Thus, the theoretical yield is \(0.00767\, \text{mol} \times 171.0\, \text{g/mol} = 1.31\, \text{g} \).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and moles of a gas (\( n \)). It is expressed as \( PV = nRT \), where \( R \) is the ideal gas constant. This equation combines several earlier laws, such as Boyle's and Charles's laws, to predict the behavior of an ideal gas.
To use the ideal gas law, remember that temperature needs to be in Kelvin, so you must convert from Celsius by adding 273.15. Pressure is often converted to atmospheres if it’s given in other units, like mmHg. The constant \( R = 0.0821 \) L atm/mol K matches these units.
The ideal gas law assumes no forces between gas molecules and that their volumes are negligible, making it useful for real gases at standard conditions—though deviations occur at high pressures or low temperatures.

Limiting Reactant

In a chemical reaction, the limiting reactant is the substance that is completely consumed first and thus determines the amount of product formed. It "limits" the reaction because once it's gone, no more product can be created. To identify the limiting reactant, compare the mole ratio of the reactants used in the balanced equation with the moles available.
For example, in the reaction \[ \mathrm{Ni}_{(s)} + 4\mathrm{CO}_{(g)} \rightarrow \mathrm{Ni}(\mathrm{CO})_4 \], one mole of Nickel (\( \mathrm{Ni} \)) reacts with four moles of Carbon Monoxide (\( \mathrm{CO} \)). Having calculated the moles of both \( \mathrm{Ni} \) and \( \mathrm{CO} \), you can determine which reactant is consumed first.
If calculations show more \( \mathrm{CO} \) available than required, \( \mathrm{Ni} \) becomes the limiting reactant. The limiting reactant concept is crucial for calculating the theoretical yield of a reaction.

Chemical Equations

Chemical equations are symbolic representations of chemical reactions. They show the reactants, products, and their ratios. A balanced chemical equation has equal numbers of each type of atom on both sides, which follows the law of conservation of mass.
Balancing involves ensuring that the same quantity of each element appears in the reactants and the products. For example, in the equation for the formation of Nickel Carbonyl (\( \mathrm{Ni}(\mathrm{CO})_4 \)): \[ \mathrm{Ni} + 4\mathrm{CO} \rightarrow \mathrm{Ni}(\mathrm{CO})_4 \],
one atom of Nickel and four molecules of Carbon Monoxide combine to create one molecule of \( \mathrm{Ni}(\mathrm{CO})_4 \).
Understanding how to balance chemical equations is essential, as it allows chemists to accurately predict the amounts of substances consumed and produced in reactions. It also plays a critical role in stoichiometric calculations and reactions involving limiting reactants.