Question

A compound has the empirical formula \(\mathrm{CHCI}\) A \(256-\mathrm{mL}\) flask, at \(373 \mathrm{K}\) and \(750 .\) torr, contains \(0.800 \mathrm{g}\) of the gaseous compound. Give the molecular formula.

Short answer

The molecular formula of the gaseous compound is C\(_2\)H\(_2\)Cl\(_2\).

Step-by-step solution

Convert the given pressure to atm.

To use the ideal gas equation, we need the pressure in atmospheres (atm). Convert the given pressure (750 torr) to atm: 1 atm = 760 torr \(Pressure = \frac{750 \, torr}{760 \, \frac{torr}{atm}} = 0.9868 \, atm\)

Calculate the moles of the gaseous compound.

Use the ideal gas equation \(PV = nRT\) to calculate the moles (n) of the compound: Rearrange the equation to solve for n: \(n = \frac{PV}{RT}\) Substitute the given values and constants (R = 0.0821 \(L \cdot atm \cdot K^{-1} \cdot mol^{-1}\)): \(n = \frac{(0.9868 \, atm)(0.256 \, L)}{(0.0821 \, L \cdot atm \cdot K^{-1} \cdot mol^{-1})(373 \, K)} = 0.008962 \, mol\)

Determine the molar mass of the gaseous compound.

Divide the mass of the compound (0.800 g) by the moles (0.008962 mol) to get the molar mass of the gaseous compound: Molar mass = \(\frac{0.800 \, g}{0.008962 \, mol} = 89.25 \, \frac{g}{mol}\)

Calculate the empirical formula mass.

Calculate the mass of the empirical formula CHCI: C: 12.01 g/mol H: 1.008 g/mol Cl: 35.45 g/mol Empirical formula mass = 12.01 g/mol + 1.008 g/mol + 35.45 g/mol = 48.468 g/mol

Determine the ratio between the molar mass and empirical formula mass.

Divide the molar mass of the compound (89.25 g/mol) by the empirical formula mass (48.468 g/mol) to get the ratio: Ratio = \(\frac{89.25 \, \frac{g}{mol}}{48.468 \, \frac{g}{mol}} = 1.841\)

Determine the molecular formula.

Because the ratio between the molar mass and empirical formula mass is close to 2 (1.841), the molecular formula will have twice the amount of elements present in the empirical formula. Molecular Formula = (CHCI)\(_2\) = C\(_2\)H\(_2\)Cl\(_2\) The molecular formula of the gaseous compound is C\(_2\)H\(_2\)Cl\(_2\).

Key concepts

Empirical Formula

The empirical formula of a compound is the simplest ratio of atoms of each element present in the compound. It doesn't reflect the actual number of atoms, but rather the smallest whole number ratio that depicts the proportion of different elements.

For example, ethylene (C₂H₄) and benzene (C₆H₆) both have the same empirical formula, CH, because the ratio of carbon to hydrogen is the same when reduced to the simplest whole numbers. In the case of the exercise, the empirical formula CHCl represents the simplest ratio of the elements present in the gaseous compound.

Understanding the empirical formula is crucial because it serves as the foundation for determining the actual molecular formula, which gives us the precise number of atoms of each element in a molecule. It is often the first step in identifying an unknown compound during chemical analysis.

Molar Mass Calculation

Molar mass, often expressed in grams per mole (g/mol), is a fundamental concept in chemistry that refers to the mass of one mole of a substance. One mole of any substance contains Avogadro's number of entities (approximately 6.022 x 10²³), whether they are atoms, molecules, ions, or other particles.

To calculate the molar mass of a compound, add up the masses of each individual element in the compound based on their atomic weights and the number of times each element occurs in the molecular formula. In the presented exercise, the molar mass calculation is critical to determine the actual molecular formula from the empirical formula.

For the gaseous compound with the given empirical formula CHCl, the molar mass is calculated by dividing the mass given in the problem (0.800 g) by the number of moles (0.008962 mol). This step is essential for comparing the empirical formula mass to the molar mass of the compound to find the ratio necessary for determining the molecular formula.

Ideal Gas Law

The ideal gas law is a crucial equation in the study of thermodynamics and chemistry. The law is a mathematical relationship that combines several previous laws about gases into one formula: PV = nRT; where P stands for pressure, V for volume, n for moles of gas, R for the ideal gas constant, and T for temperature.

In this exercise, the ideal gas law allows us to calculate the number of moles of the gaseous compound under specific temperature and pressure conditions. The temperature and volume are given, while the gas constant (R) is known (0.0821 L·atm·K^-1·mol^-1). First, we convert the pressure from torr to atm to match the units used in the gas constant. Then, by rearranging the ideal gas equation to solve for n (moles), we input the given values to find the amount of gas in moles.

As an exercise improvement advice, remember that real gases do not always follow the ideal gas law perfectly, especially under conditions of high pressure or low temperature, due to intermolecular forces and the volume occupied by the gas molecules themselves. However, for the purpose of this problem, we assume that the compound behaves as an ideal gas.