Question

A compound has the empirical formula \(\mathrm{SF}_{4}\). At \(20^{\circ} \mathrm{C}\), \(0.100 \mathrm{~g}\) of the gaseous compound occupies a volume of \(22.1 \mathrm{~mL}\) and exerts a pressure of \(1.02 \mathrm{~atm} .\) What is the molecular formula of the gas?

Short answer

S2F8

Step-by-step solution

Use the Ideal Gas Law

To find the molecular formula, determine the molar mass using the ideal gas law, which is expressed as \[PV = nRT\].Where: - \(P\) is the pressure (1.02 atm),- \(V\) is the volume (0.0221 L by converting 22.1 mL to L),- \(R\) is the ideal gas constant \(0.0821 \, \mathrm{L.atm/mol.K}\),- \(T\) is the temperature in Kelvin (293 K, converted from 20°C).Let's solve for \(n\), the number of moles: \[n = \frac{PV}{RT} = \frac{1.02 \times 0.0221}{0.0821 \times 293}\].

Key concepts

Empirical formula

An empirical formula represents the simplest whole-number ratio of atoms of each element in a compound. It gives an idea about the constituent elements and their proportions but not the actual number of atoms in the molecules. In our example, the compound with the empirical formula \(\mathrm{SF}_4\) suggests that sulfur and fluorine are present in a 1:4 ratio. However, it doesn't reveal the actual count of these atoms in each mole of the compound. To identify the actual molecular formula, additional information such as molar mass is needed.

It's important to remember that the empirical formula is often a starting point for determining the molecular formula, as both can sometimes be identical or multiples of each other. Thus, understanding the empirical formula is a foundational step towards knowing the exact molecular composition of a compound.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is represented by the equation \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) represents volume, \(n\) is the number of moles, \(R\) is the ideal gas constant (0.0821 L.atm/mol.K), and \(T\) is the temperature in Kelvin.

This equation is invaluable for calculating any one of these properties if the others are known. To apply this in our problem, pressure (1.02 atm), volume (0.0221 L from 22.1 mL), and temperature (293 K from 20°C) are used to compute the number of moles \(n\). By rearranging the formula, the value of \(n\) is obtained, aiding in further calculations like finding the molar mass. The Ideal Gas Law is key especially for gaseous substances and deviates slightly only under high pressure or low temperature.

Molar mass calculation

Molar mass calculation is crucial for determining the molecular formula once the empirical formula is known. The mass of one mole of a substance (usually in g/mol) allows for the transition from the empirical to the molecular formula. In the exercise, after using the Ideal Gas Law (\(PV = nRT\)) to find the number of moles, you can calculate the molar mass by dividing the mass of the known sample (0.100 g) by the number of moles obtained.

Mathematically, it is expressed as follows:\[\text{Molar Mass} = \frac{\text{mass of sample}}{n}\]
By comparing the calculated molar mass to the molar mass suggested by the empirical formula, you can determine whether the molecular formula is a multiple of the empirical formula. In cases where the calculated molar mass is the same as the empirical molar mass, the two formulas are identical.

Pressure conversion

Pressure conversion is often necessary in chemical calculations, especially when dealing with gases. Because pressure can be measured in various units such as atmospheres, pascals, or mmHg, conversions ensure consistency and correctness in calculations. In our problem, pressure is given in atmospheres (atm), which is suitable for the Ideal Gas Law, since its constant \(R\) usually operates with atm.If, however, your problem involved a different pressure unit, a conversion would be required to match the units.Common conversion factors include:

• 1 atm = 101.3 kPa (kilopascals)
• 1 atm = 760 mmHg (or torr)
• 1 mmHg = 133.322 Pa (pascals)

These conversions facilitate solving gas-related problems effortlessly.