Question

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.

Short answer

The most dense gas at 1.00 atm and 298 K among CO₂, N₂O, and Cl₂ is Cl₂ with a density of 2.90 g/L.

Step-by-step solution

Write down the Ideal Gas Law

Recall the Ideal Gas Law equation: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Rearrange Ideal Gas Law for molar volume

We want to find density, which is mass per unit volume. So first, let's find the volume for each gas per mole. Rearrange the Ideal Gas Law equation for molar volume (V/n) and solve for it: \[\frac{V}{n} = \frac{RT}{P}\] Now, we can plug in the given temperature, T = 298 K, and pressure, P = 1.00 atm, and the value of R = 0.0821 L atm K⁻¹ mol⁻¹: \[\frac{V}{n} = \frac{(0.0821 \,\mathrm{L\,atm\,K^{-1}\,mol^{-1}})(298 \,\mathrm{K})}{1.00\, \mathrm{atm}}\]

Calculate molar volume

Simplify the expression to find the molar volume: \[\frac{V}{n} = 24.45\, \mathrm{L/mol}\]

Calculate densities of the gases

Now that we have the molar volume, we can calculate the density for each gas by dividing its molar mass by the molar volume. The molar masses of CO₂, N₂O, and Cl₂ are 44.01 g/mol, 44.02 g/mol, and 70.9 g/mol, respectively. For CO₂: \[\rho_{CO_{2}} = \frac{44.01 \, \mathrm{g/mol}}{24.45 \, \mathrm{L/mol}} = 1.80 \, \mathrm{g/L}\] For N₂O: \[\rho_{N_{2}O} = \frac{44.02 \, \mathrm{g/mol}}{24.45 \, \mathrm{L/mol}} = 1.80 \, \mathrm{g/L}\] For Cl₂: \[\rho_{Cl_{2}} = \frac{70.9 \, \mathrm{g/mol}}{24.45 \, \mathrm{L/mol}} = 2.90 \, \mathrm{g/L}\]

Compare the densities

Compare the calculated densities of CO₂, N₂O, and Cl₂: \(\rho_{CO_{2}} = 1.80 \, \mathrm{g/L}\) \(\rho_{N_{2}O} = 1.80 \, \mathrm{g/L}\) \(\rho_{Cl_{2}} = 2.90 \, \mathrm{g/L}\)

Determine the most dense gas

Since Cl₂ has the highest density (2.90 g/L) at the given conditions (1.00 atm and 298 K), it is the most dense gas among CO₂, N₂O, and Cl₂.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that allows us to predict and understand the behavior of gases under various conditions. At the heart of this concept is the equation: \[PV = nRT\],where P represents the pressure, V stands for volume, n is the number of moles of the gas, R is the universal gas constant, and T denotes the temperature in Kelvin. The beauty of the Ideal Gas Law lies in its ability to model the state of an 'ideal' gas—a theoretical gas that perfectly follows the law's assumptions of internal energy depending only on temperature and no interactions between the gas particles aside from perfectly elastic collisions.
When dealing with real gases, this law can still provide reasonable approximations under many conditions, such as standard temperature and pressure. By manipulating this equation, scientists and students alike can explore various properties, such as the molar volume—that is, the volume one mole of the gas occupies. In the given exercise, using this law provided the foundation for solving the density question, asserting its indispensable role in the study of gaseous substances.

Gas Density

Gas density, which represents the mass of a gas per unit volume, plays an instrumental role in determining how substances behave under different environmental conditions. It's crucial for many applications, including industrial processes and environmental science. Unlike liquids and solids, the density of gases is highly dependent on pressure and temperature. To find the density of a gas, the equation \[\rho = \frac{m}{V}\] is often used, where \(\rho\) is the density, m is the mass, and V is the volume.
In the context of the exercise, the molar volume obtained from the Ideal Gas Law was used to calculate the density of different gases, allowing for a direct comparison. Since all gases in the problem are at the same temperature and pressure, the gas with the greatest molar mass displayed the highest density. This fact illustrates the direct relationship between gas density and molar mass, making density a derived property that hinges not only on the environmental conditions but also on the intrinsic characteristics of the gas itself.

Molar Mass

Molar mass is the weight of 1 mole (6.022 x 10²³ particles) of any chemical substances, expressed in grams per mole (g/mol). It's a cardinal quantity in chemistry because it bridges the microscopic world of atoms and molecules with the macroscopic world we interact with every day. Knowing the molar mass aids in converting between moles, which count particles, and grams, which measure mass. This is a pivotal step in chemical calculations for creating solutions, stoichiometry, and predicting yields.
In our exercise, the molar mass directly influenced the density calculations for each gas. Since the gases shared the same molar volume under the given conditions, it was simply a matter of dividing their molar mass by this common volume to find their respective densities. Here we can see that the molar mass forms the crux of determining a substance's density when the volume is standardized. This demonstration of molar mass influencing gas density is a vivid portrayal of the intrinsic link between these two critical concepts in chemistry.