Which gas is most dense at \(1.00 \mathrm{~atm}\) and \(298 \mathrm{~K}: \mathrm{CO}_{2}, \mathrm{~N}_{2} \mathrm{O}\), or \(\mathrm{Cl}_{2}\) ? Explain.

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas under various conditions. The law is expressed by the formula:
\[PV = nRT\]
where \(P\) represents pressure, \(V\) denotes volume, \(n\) is the number of moles of gas, \(R\) stands for the ideal gas constant, and \(T\) is the absolute temperature in kelvins. For calculation purposes, it is crucial to use the ideal gas constant \(R = 0.08206 \text{L.atm/mol.K}\).
What makes the Ideal Gas Law so valuable is its ability to predict how a gas will behave under different conditions of temperature, pressure, and volume. It's based on the assumptions that ideal gas particles do not interact with each other and occupy no space — although no gas perfectly fits this description, the law provides a good approximation for many gases under normal conditions.
When combining the Ideal Gas Law with other concepts such as density, we can perform more complex calculations, like finding the density of a gas at specific temperature and pressure conditions.

Molar mass serves as a bridge in many chemical calculations, linking grams to moles, an essential conversion in chemistry. It is defined as the mass of one mole of a substance and expressed in grams per mole (g/mol). One mole of any substance contains Avogadro's number of entities (e.g., atoms, molecules), which is approximately \(6.022 \times 10^{23}\) entities.
In gas density calculations, the molar mass is used to convert the number of moles to mass. Since the Ideal Gas Law includes the variable \(n\) for moles, we can rewrite the equation to find the density \(\rho\) by multiplying the number of moles by the molar mass and dividing by the volume:
\[\rho = \frac{mass}{volume} = \frac{n \times M}{V}\]
Understanding the molar mass is crucial as it helps us identify how much one mole of gas weighs and allows us to compare the densities of different gases once combined with the Ideal Gas Law equation.

The molar volume of a gas is the volume occupied by one mole of the gas at a given temperature and pressure. According to the Ideal Gas Law, one mole of any ideal gas occupies 22.4 liters at standard temperature and pressure (STP), which is 0°C (273 K) and 1 atmosphere (atm) of pressure. However, for different conditions of temperature and pressure, the molar volume will change.
In our exercise, we are not using the standard conditions, so the volume is not 22.4 liters. Instead, we obtain the volume variable by isolating it from the Ideal Gas Law expression. But for density calculations, we are usually more interested in the mass to volume ratio, and that’s where the concept of molar volume gets replaced by the calculation of the gas's density at the specific conditions provided.

Comparing gas densities is a process that involves using the Ideal Gas Law and the concept of molar mass to determine which gas is denser under certain conditions. To compare the densities of gases, as in our textbook problem, we rearrange the Ideal Gas Law to express density (\(\rho\)):
\[\rho = \frac{PM}{RT}\]
Using this formula, we calculate the density of each gas individually by substituting the known values, such as the gas's molar mass, the temperature, the pressure, and the ideal gas constant, into the equation. Finally, after computing the values, we compare the densities directly.

• The higher the molar mass, the higher the density at a given temperature and pressure.
• The higher the temperature, with other variables held constant, the lower the density.
• The higher the pressure, with other variables held constant, the higher the density.

In this exercise, since the temperature and pressure are the same for all gases, the gas with the greatest molar mass, \(Cl_2\), is the most dense. This direct comparison method is straightforward and highly effective for understanding the relative density of gases in different scenarios.