Question

Nickel carbonyl, \(\mathrm{Ni}(\mathrm{CO})_{4},\) can be made by the roomtemperature reaction of finely divided nickel metal with gaseous CO. This is the basis for purifying nickel on an industrial scale. If you have CO in a sealed \(1.50-\mathrm{L}\) flask at a pressure of \(418 \mathrm{mmHg}\) at \(25.0^{\circ} \mathrm{C},\) calculate the maximum mass in grams of \(\mathrm{Ni}(\mathrm{CO})_{4}\) that can be made.

Short answer

The maximum mass of Ni(CO)₄ that can be made is approximately 1.43 g.

Step-by-step solution

Convert Pressure to Atm

First, we need to convert the pressure of CO from mmHg to atm since standard units are easier to work with. We use the conversion factor: \(1 \text{ atm} = 760 \text{ mmHg}\).\[P_{\text{CO}} = \frac{418 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.550 \text{ atm}\]

Use Ideal Gas Law to Find Moles of CO

We use the ideal gas law equation, \(PV = nRT\), to find the number of moles of CO. \(R = 0.0821 \frac{L \cdot atm}{mol \cdot K}\). The temperature in Kelvin is \(25.0 + 273.15 = 298.15 \text{ K}\). Solve for \(n\): \[n = \frac{PV}{RT} = \frac{(0.550 \text{ atm})(1.50 \text{ L})}{(0.0821 \frac{L \cdot atm}{mol \cdot K})(298.15 \text{ K})} \approx 0.0336 \text{ mol}\]

Calculate Mass of Nickel Carbonyl

Since the reaction \(\text{Ni} + 4\text{CO} \rightarrow \text{Ni(CO)}_4\) shows a 1:4 stoichiometric ratio between Ni(CO)\(_4\) and CO, we can make \(0.0336 \text{ mol} / 4 = 0.0084 \text{ mol}\) of Ni(CO)\(_4\). The molar mass of Ni(CO)\(_4\) is \(58.69 + 4(12.01 + 16.00) = 170.69 \text{ g/mol}\). Calculate the mass:\[m = 0.0084 \text{ mol} \times 170.69 \text{ g/mol} \approx 1.43 \text{ g}\]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant, and \(T\) for temperature in Kelvin. This equation is pivotal when calculating unknown properties of a gas given the other variables.
When you're given the pressure in mmHg, as in the problem with carbon monoxide (CO), you'll typically need it in atmospheres for the ideal gas law. Hence, converting the units is crucial.
After conversion, you'll insert these values to compute the number of moles of the gas. Since the ideal gas constant \(R\) is \(0.0821 \frac{L\cdot atm}{mol\cdot K}\), ensure all units align correctly: pressure in atm, volume in liters, and temperature in Kelvin.
This powerful tool in chemistry simplifies the calculations regarding gases, making it straightforward to identify quantities like the number of moles, which is then used in further stoichiometric calculations.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. In our example, determining how much nickel carbonyl \(\text{Ni(CO)}_4\) can be produced requires using stoichiometric ratios derived from the chemical equation \(\text{Ni} + 4\text{CO} \rightarrow \text{Ni(CO)}_4\).
These ratios are crucial: the equation shows that one mole of nickel carbonyl is formed from four moles of carbon monoxide. This knowledge allows us to relate moles of CO to moles of \(\text{Ni(CO)}_4\), which we found from the Ideal Gas Law calculation. Hence, understanding this ratio helps calculate the number of moles of \(\text{Ni(CO)}_4\) that can be produced from the available moles of CO.
After determining the moles of \(\text{Ni(CO)}_4\), we use the molar mass for converting these moles to mass, utilizing \(58.69 + 4(12.01 + 16.00) = 170.69 \text{ g/mol}\). Without stoichiometry, it would be challenging to predict the amounts of products in a reaction using known quantities of reactants.

Gas Pressure Conversion

Converting gas pressure to different units is often necessary when working with gases, especially when using the Ideal Gas Law. In our problem, the pressure of CO was initially given in mmHg but needed to be in atm for uniformity and to fit the \(PV = nRT\) equation standards.
This conversion uses a simple factor, where \(760\, \text{mmHg} = 1\, \text{atm}\). By dividing the pressure of CO given in mmHg by 760, you obtain the pressure in atm. This step ensures that when you apply the Ideal Gas Law equation, each component uses compatible units to give meaningful results.
Bringing all units to an appropriate standard is vital for accurate calculations. Incorrect conversions or mismatched units can easily lead to errors in determining the amount of a gas, underlining why conversions are a necessary skill in chemistry.