A gaseous hydrocarbon that is \(82.7 \%\) C and \(17.3 \%\) H by mass has a density of \(2.35 \mathrm{g} / \mathrm{L}\) at \(25^{\circ} \mathrm{C}\) and 752 Torr. What is the molecular formula of this hydrocarbon?

We're given the percentages by mass of carbon (C) and hydrogen (H) in the hydrocarbon. Assume that we have 100 g of the hydrocarbon. Thus, 82.7 g would be Carbon and 17.3 g would be Hydrogen. The molar masses of Carbon and Hydrogen are approximately 12.01 g/mol and 1.008 g/mol, respectively. So, we can calculate the number of moles of each element: \[ Moles\ of\ C = \frac{82.7 g}{12.01 g/mol} \approx 6.88 mol \] \[ Moles\ of\ H = \frac{17.3 g}{1.008 g/mol} \approx 17.16 mol \] The ratio of moles of C to H is approximately \( \frac{6.88}{6.88} \) : \( \frac{17.16}{6.88} \) or 1:2.5. Round 2.5 to the nearest whole number to find that the empirical formula is \( C_{1}H_{2.5} \), which does not exist as a hydrocarbon, so we round it up to C2H5.

Next, we'll use the density and the Ideal Gas Law to find the molar mass of the hydrocarbon. The Ideal Gas Law is PV= nRT. The values we have are: P = 752 Torr, V = 1 L, R = 62.36 (Torr. L) / (mol. K), and T = 298 K (25°C+ 273). Solving for n, we get: \[ n = \frac {PV}{RT} = \frac {752 Torr * 1 L} {62.36 Torr.L/mol.K * 298 K} \approx 0.0406 mol \] The mass of 1 L of the gas is 2.35 g, which is the mass of 0.0406 moles (from the calculation above). So, the molar mass (M) of the gas can be found using the formula: \[ M = \frac {mass}{moles} = \frac {2.35 g}{0.0406 mol} \approx 58 g/mol\]

Now that we have the empirical formula and the molar mass, we can find the molecular formula. First, find the molar mass of the empirical formula \( C_{2}H_{5} \), which is about \( 2(12.01 g/mol) + 5(1.008 g/mol)= 29 g/mol \). Then, divide the molar mass of the molecule by the molar mass of the empirical formula to find the number of empirical formula units in each molecule of the hydrocarbon: \[ \frac {58 g/mol}{29 g/mol}= 2 \] Multiply the subscripts in your empirical formula by this number to get the molecular formula: \( C_{2*2}H_{5*2} = C_{4}H_{10} \)