Question

Order the following gases according to increasing density: \(\mathrm{CO} ; \mathrm{CO}_{2} ; \mathrm{H}_{2} \mathrm{~S}\). The temperature and pressure are the same for all three samples.

Short answer

The gases in order of increasing density are: CO, H2S, CO2.

Step-by-step solution

Note Given Conditions

Recognize that the temperature and pressure are the same for all three gases, which implies that the Ideal Gas Law applies to each gas equally. Thus, under the same conditions of temperature and pressure, the density of a gas is proportional to its molar mass.

Determine the Molar Mass

Calculate the molar mass of each gas. The molar mass of CO (carbon monoxide) is 28.01 g/mol, for CO2 (carbon dioxide) it is 44.01 g/mol, and for H2S (hydrogen sulfide) it is 34.08 g/mol.

Order by Increasing Molar Mass

Since density is proportional to molar mass, order gases according to their molar mass. Hence, the order from least dense to most dense is H2S, CO, CO2.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the relationship between pressure (P), volume (V), temperature (T), and the amount of gas in moles (n) using the equation \( PV = nRT \), where R is the universal gas constant. Because temperature and pressure are held constant in the exercise, this law helps us infer that, for a given amount of gas, the only variable that will affect the density is the molar mass.

The equation itself does not include density, but by rearranging it to solve for the moles (n) and then utilizing the definition of density (mass per unit volume), it's possible to relate molar mass to density. Therefore, under constant temperature and pressure conditions, comparing the density of gases boils down to comparing their molar masses, which is a direct application of the Ideal Gas Law in this context.

Molar Mass Calculation

Molar mass represents the mass of one mole of a substance, typically measured in grams per mole (\(g/mol\)). It's calculated by summing the atomic masses of all the atoms in a molecule, with these values found on the periodic table. For example, as shown in the solution, carbon monoxide (CO) has a molar mass of 28.01 g/mol, calculated by adding the atomic mass of carbon (12.01 g/mol) and oxygen (16.00 g/mol).

Similarly, carbon dioxide (\(CO_2\)) has a molar mass of 44.01 g/mol, and hydrogen sulfide (\(H_2S\)) has a molar mass of 34.08 g/mol. Knowing the molar mass of each gas allows us to predict their relative densities. Molar mass is crucial because it bridges the gap between the microscopic scale of atoms and molecules and the macroscopic world that we can measure and observe, effectively interlinking concepts across different scales in chemistry.

Comparison of Gas Properties

Comparing gas properties, especially density, under constant conditions allows us to predict how different gases will behave relative to one another. Density is defined as the mass of a substance divided by its volume (\( \text{density} = \frac{\text{mass}}{\text{volume}} \)). When discussing gases, density is directly proportional to the molar mass of the gas when the temperature and pressure are constant, as stated by the Ideal Gas Law.

Therefore, understanding how to compare the molar mass of different gases is essential to predict the relative densities. In the given exercise, by calculating the molar mass of each gas and understanding that a higher molar mass indicates a higher density under the same conditions, students can logically deduce that \(CO_2\), with the highest molar mass, will be the densest, followed by \(H_2S\), and \(CO\) will be the least dense. This comparison is an integral part of chemistry education, which helps in developing a deeper comprehension of gas behaviors and their practical implications in both laboratory and industrial contexts.