Question

A 2.650 g sample of a gaseous compound occupies \(428 \mathrm{mL}\) at \(24.3^{\circ} \mathrm{C}\) and \(742 \mathrm{mmHg} .\) The compound consists of \(15.5 \%\) C \(, 23.0 \%\) Cl, and \(61.5 \%\) F, by mass. What is its molecular formula?

Short answer

The molecular formula of the compound is \(C_4Cl_2F_{10}\).

Step-by-step solution

Calculate the molar mass of the compound

The molar mass can be calculated using the ideal gas law \(PV=nRT\) where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in kelvins. Rewriting this equation for n, we get \(n=PV/RT\). First, convert temperature from Celsius to Kelvins by adding 273. Then, convert the pressure from mmHg to atm by dividing it by 760. Finally, calculate n by substituting P \(=742/760=0.976\) atm, V=0.428L, R=0.0821 L.atm/mol.K, and T=24.3+273=297.3K into the equation. This gives n=\(0.976*0.428/0.0821*297.3=0.011\) moles. The molar mass is then given by the ratio mass/n, which is \(2.650/0.011=240.9\) g/mol.

Calculate the empirical formula

Calculate the mass of each element in the compound using the given percentages. This yields 0.410 g C, 0.6095 g Cl, and 1.630 g F. Divide these masses by their atomic masses to get the number of moles of each element: \(0.410/12.01=0.0341\) moles C, \(0.6095/35.45=0.0172\) moles Cl, and \(1.630/18.998=0.0857\) moles F. Divide these numbers of moles by the smallest one to get the ratio of atoms in the empirical formula. This gives C2ClF5. So the empirical formula is \(C_2ClF_5\).

Determine the molecular formula

The molecular formula is a multiple of the empirical formula. Calculate the molar mass of the empirical formula, which is \(2*12.01+35.45+5*18.998=120.45\) g/mol. Divide the molar mass of the compound by the molar mass of the empirical formula to get the multiple, which is \(240.9/120.45=2\). So the molecular formula is \(C_4Cl_2F_{10}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a cornerstone concept in chemistry, bridging the gap between the macroscopic and microscopic worlds of gases. It is expressed as the equation \(PV = nRT\), where:

• \(P\) is the pressure of the gas, measured in atmospheres (atm) for standard conditions.
• \(V\) is the volume of the gas, typically measured in liters (L).
• \(n\) is the number of moles of gas, representing the quantity of substance.
• \(R\) is the ideal gas constant, equal to \(0.0821 \, \text{L.atm/mol.K}\).
• \(T\) is the temperature, measured in Kelvin (K).

In practical applications, we often use this equation to find the amount of gas in moles \(n\), which helps in further calculations, such as determining molar masses or verifying purity. To use the Ideal Gas Law effectively, one must first standardize the units:

• Convert pressures from mmHg to atm by dividing by 760.
• Convert temperatures from Celsius to Kelvin by adding 273.15.

Following these conversions ensures consistent calculations and accurate results.

Empirical Formula

The empirical formula is the simplest integer ratio of the elements present in a compound. It is not necessarily unique to the compound but represents the proportion of atoms in the smallest whole numbers.To determine the empirical formula in exercises such as this, follow these steps:

• Identify the mass percentage of each element in the compound.
• Convert these percentages to grams, assuming you start with a 100 grams of substance, simplifying calculations.
• Divide each element's mass by its atomic mass to find the number of moles of each element.
• Identify the smallest number of moles calculated and normalize each element's mole count by dividing the others by this value, giving the simplest whole number ratio.
• These ratios provide the subscripts for each element in the empirical formula.

In the example provided, the empirical ratios were found for carbon, chlorine, and fluorine, leading to the formula \(C_2ClF_5\). This foundational step is crucial for moving on to determine the molecular formula.

Molecular Mass Calculation

Molecular mass, also known as molecular weight, is the sum of the atomic masses of all atoms in a molecule. It's a more detailed measure than the empirical formula provides, reflecting the exact number of atoms of each type in a molecule.To calculate the molecular mass:

• Determine the empirical formula mass, which involves adding up the atomic masses as dictated by the empirical formula.
• Compare the known molar mass of the compound determined via experimental data and the Ideal Gas Law calculation to the empirical formula mass.
• Divide the molar mass by the empirical formula mass to find a multiplier.
• Apply this multiplier to each subscript in the empirical formula to derive the molecular formula.

In this case, the empirical formula \(C_2ClF_5\) gave a mass of \(120.45 \, \text{g/mol}\). Given the experimentally derived molecular mass of \(240.9 \, \text{g/mol}\), the ratio was \(2\). Thus, the molecular formula \(C_4Cl_2F_{10}\) represents the true composition of the compound, precisely doubling each element's presence in the empirical formula.