Which of these has the greatest density? (a) \(\mathrm{SF}_{6}\) at \(\mathrm{STP}\) (b) \(\mathrm{C}_{2} \mathrm{H}_{6}\) at room conditions (c) \(\mathrm{He}\) at \(-80^{\circ} \mathrm{C}\) and \(2.15\) atm

The Ideal Gas Law is a fundamental principle in chemistry that allows us to relate the pressure (\(P\text{, in atm}\)), volume (\(V\text{, in liters}\)), quantity of gas (\(n\text{, in moles}\)), and temperature (\(T\text{, in Kelvins}\)) of a gas sample. It is concisely represented by the equation:
\[PV = nRT\]
where \(R\) stands for the gas constant, having the value \(0.0821 \text{ L·atm/(mol·K)}\). To solve problems involving this law, we must ensure we're using the correct units: pressure in atmospheres, volume in liters, number of moles, and temperature in Kelvins.

For example, if we want to determine the volume occupied by helium (\(He\text{ gas}\)) at a certain temperature and pressure, we simply plug in our values into the Ideal Gas Law and solve for the volume. This law is crucial for the understanding of how gases behave under different conditions, and assists in making predictions about their properties, such as density, which can be calculated by dividing the gas’s molar mass by the volume it occupies.

Molar mass is the weight of one mole of a chemical element or compound. It reflects the combined mass of all atoms in the molecule and is typically expressed in grams per mole (g/mol). The molar mass is critical in comparing the density of gases, as it provides the mass component in the density equation \(\rho = \frac{m}{V}\).

To calculate molar mass, add up the atomic masses of all atoms in the molecule. For example, sulfur hexafluoride (\(SF_6\text{ gas}\)) has a molar mass of 146.06 g/mol, calculated by summing the atomic masses of one sulfur and six fluorine atoms. Understanding molar mass allows us to determine how much of a particular gas can be found in a specific volume, a necessary step in comparing gas densities under various conditions.

Standard Temperature and Pressure (STP) are reference points used to report the properties of materials. By convention, STP is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (atm). At STP, one mole of an ideal gas occupies 22.4 liters, known as the molar volume of a gas.

The conditions of STP are essential for comparing the properties of different substances under consistent conditions. According to Avogadro's Law, equal volumes of gases at the same temperature and pressure contain the same number of particles (or moles). Consequently, if gases like \(SF_6\) and \(C_2H_6\) are both at STP, they will occupy the same volume despite their different molar masses. This makes STP a valuable comparison point when determining the density of gases, as it eliminates the volume variable from the density equation, thus allowing for a direct comparison based on molar mass alone.