Problem 26 - Chemistry: Principles and Reactions

Question

Phosgene is a highly toxic gas made up of carbon, oxygen, and chlorine atoms. Its density at \(1.05\) atm and \(25^{\circ} \mathrm{C}\) is \(4.24 \mathrm{~g} / \mathrm{L}\). (a) What is the molar mass of phosgene? (b) Phosgene is made up of \(12.1 \% \mathrm{C}, 16.2 \% \mathrm{O}\), and \(71.7 \% \mathrm{Cl}\). What is the molecular formula of phosgene?

Step-by-Step Solution

Answer

Question: Determine the molar mass and molecular formula of phosgene, given its density is 4.24 g/L at 25°C and 1.05 atm, and its percentage composition by mass is 12.1% C, 16.2% O, and 71.7% Cl. Answer: The molar mass of phosgene is approximately 98.91 g/mol, and its molecular formula is COCl₂.

Step 1: Use the Ideal Gas Law to find the molar mass of phosgene.

The Ideal Gas Law is given by \(PV=nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant and \(T\) is the temperature. We can rearrange the formula to find the molar mass of phosgene: \(PV=nRT \Rightarrow P=\frac{nRT}{V}\) Since the molar mass \(M\) is the ratio between mass \(m\) and number of moles \(n\), \(M=\frac{m}{n}\), we can rewrite the equation in terms of mass and molar mass as: \(P=\frac{m}{M}\frac{RT}{V} \Rightarrow M=\frac{mRT}{PV}\) Now, we plug in the given values: \(M=\frac{(4.24\times 0.0821\times (273+25))}{(1.05\times 1)}\)

Step 2: Calculate the molar mass of phosgene

From the previous step, we can now calculate the molar mass: \(M=\frac{(4.24\times 0.0821\times 298)}{1.05} \approx 98.91 \, \mathrm{g/mol}\) So, the molar mass of phosgene is approximately \(98.91\, \mathrm{g/mol}\).

Step 3: Use the percentage composition by mass to find the molecular formula of phosgene

We are given the percentages of C, O, and Cl in phosgene. First, we convert these percentages into grams assuming \(100 \, \mathrm{g}\) sample of phosgene: - Mass of C = \(12.1\% \times 100 = 12.1 \, \mathrm{g}\) - Mass of O = \(16.2\% \times 100 = 16.2 \, \mathrm{g}\) - Mass of Cl = \(71.7\% \times 100 = 71.7 \, \mathrm{g}\) Then, we divide each mass by the corresponding molar mass of each element to find the mole ratio: - Moles of C = \(\frac{12.1}{12.01} \approx 1.01\) - Moles of O = \(\frac{16.2}{16.00} \approx 1.01\) - Moles of Cl = \(\frac{71.7}{35.45} \approx 2.02\) Finally, we divide each mole ratio by the smallest mole ratio to get the molecular formula: - Ratio of C:O:Cl = \(\frac{1.01}{1.01} : \frac{1.01}{1.01} : \frac{2.02}{1.01} = 1:1:2\) So, the molecular formula of phosgene is \(\mathrm{COCl_2}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a valuable principle in chemistry that allows us to relate the pressure, volume, temperature, and number of moles of a gas. This law is represented by the formula: \(PV = nRT\), where:

• \(P\) is the pressure of the gas in atmospheres (atm).
• \(V\) is the volume of the gas in liters (L).
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant \(0.0821 \text{ L atm/mol K}\).
• \(T\) is the temperature in Kelvin (K).

To solve for the molar mass, which is the mass of one mole of a substance, we can rearrange the Ideal Gas Law to \(M = \frac{mRT}{PV}\). Here, \(m\) is the mass of the gas. Using the values provided in the exercise, this formula helps us determine the molar mass of phosgene by considering how much of the gas weighs in a specific volume under known conditions of temperature and pressure.

Molar Mass Calculation

The calculation of molar mass involves understanding the concept of the amount of a substance in terms of moles. Given the weight of the gas per liter and the conditions under which it is measured, we can use the rearranged Ideal Gas Law formula:\[M = \frac{mRT}{PV}\]By plugging in given values like the density \(4.24 \text{ g/L}\), the pressure \(1.05 \text{ atm}\), and the temperature \(25^{\circ}\text{C}\), converted to Kelvin \((298 \text{ K})\), we find the molar mass. It simplifies the process to solve for \(M\), resulting in approximately \(98.91 \text{ g/mol}\). This result indicates the weight of one mole of phosgene gas.

Molecular Formula Determination

The molecular formula gives the exact number of different atoms in a molecule. For phosgene, its molecular formula can be derived from its percentage composition by mass. Given:

• \(12.1\%\) carbon (C),
• \(16.2\%\) oxygen (O), and
• \(71.7\%\) chlorine (Cl),

we assume a \(100 \text{ g}\) sample for simplicity. Thus:

• \(12.1 \text{ g}\) of carbon,
• \(16.2 \text{ g}\) of oxygen, and
• \(71.7 \text{ g}\) of chlorine.

To find the moles of each element, divide their respective masses by their atomic masses: \(12.01 \text{ g/mol}\) for carbon, \(16.00 \text{ g/mol}\) for oxygen, and \(35.45 \text{ g/mol}\) for chlorine. Calculating these gives you:

• \(\frac{12.1}{12.01} \approx 1.01\) moles of carbon,
• \(\frac{16.2}{16.00} \approx 1.01\) moles of oxygen,
• \(\frac{71.7}{35.45} \approx 2.02\) moles of chlorine.

The ratios \(1:1:2\) correspond to the atoms in the molecular formula \(\text{COCl}_2\), confirming the formula tells us exactly how these elements are combined in phosgene.