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?

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: \(M=\frac{(4.24\times 0.0821\times (273+25))}{(1.05\times 1)}\)

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 \(98.91\, \mathrm{g/mol}\).

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 \(\mathrm{COCl_2}\).