Question

Rank the following gases from least dense to most dense at \(101.33 \mathrm{kPa}\) and \(298 \mathrm{~K}: \mathrm{O}_{2}, \mathrm{Ar}, \mathrm{NH}_{3}, \mathrm{HCl}\).

Short answer

The ranking of gases from least dense to most dense at 101.33 kPa and 298 K is: NH₃ (0.706 kg/m³) < O₂ (1.331 kg/m³) < HCl (1.517 kg/m³) < Ar (1.661 kg/m³).

Step-by-step solution

Find the molar mass of each gas

To rank the gases based on their densities, we must first determine their molar masses. Use the periodic table to find the atomic masses of each element in a molecule. - For O₂: Molar mass = 2 × 16.00 g/mol = 32.00 g/mol - For Ar: Molar mass = 39.95 g/mol (since it's a noble gas) - For NH₃: Molar mass = 14.01 g/mol (N) + 3 × 1.01 g/mol (H) = 17.03 g/mol - For HCl: Molar mass = 1.01 g/mol (H) + 35.45 g/mol (Cl) = 36.46 g/mol

Use the ideal gas law to find density

Now we will use the ideal gas law equation to find the density for each gas under the given conditions (pressure = 101.33 kPa and temperature = 298 K). The ideal gas law is given by: \(PV = nRT\) Where: - P = pressure - V = volume - n = number of moles - R = ideal gas constant (8.314 J/mol·K) - T = temperature The density of a gas is given by: \(\rho = \frac{mass}{volume} = \frac{molar~mass~ ×~ number~of~ moles}{volume}\) Combining the ideal gas law and the density formula, we get: \(\rho = \frac{molar~mass~ ×~ P}{R ~×~ T}\) Now plug in the values for pressure (101.33 kPa), temperature (298 K), and molar mass for each gas to find their respective densities.

Calculate density for each gas

Use the formula from the previous step to calculate the densities: - For O₂: \(\rho = \frac{32.00~ g/mol × 101.33~ kPa}{8.314~ J/mol·K × 298~ K} = 1.331~ kg/m³\) - For Ar: \(\rho = \frac{39.95~ g/mol × 101.33~ kPa}{8.314~ J/mol·K × 298~ K} = 1.661~ kg/m³\) - For NH₃: \(\rho = \frac{17.03~ g/mol × 101.33~ kPa}{8.314~ J/mol·K × 298~ K} = 0.706~ kg/m³\) - For HCl: \(\rho = \frac{36.46~ g/mol × 101.33~ kPa}{8.314~ J/mol·K × 298~ K} = 1.517~ kg/m³\)

Rank the gases by density

Now that we have calculated the densities for each gas, we can rank them from least dense to most dense: 1. NH₃ (0.706 kg/m³) 2. O₂ (1.331 kg/m³) 3. HCl (1.517 kg/m³) 4. Ar (1.661 kg/m³) Thus, the ranking of gases from least dense to most dense at 101.33 kPa and 298 K is: NH₃ < O₂ < HCl < Ar.

Key concepts

Ideal Gas Law

The Ideal Gas Law is pivotal in understanding the behavior of gases under different conditions. It is represented by the equation \(PV = nRT\), where:

• \(P\) stands for pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant, approximately 8.314 J/mol·K
• \(T\) represents temperature in Kelvin

This formula combines several laws, including Boyle’s Law and Charles’s Law. The Ideal Gas Law allows us to predict how a gas will react when we change its pressure, volume, or temperature.
In density calculations, this law is particularly useful. Combining it with the density formula, \(\rho = \frac{mass}{volume}\), gives us a way to express gas density as \(\rho = \frac{Molar~Mass \times P}{RT}\). This highlights how gas density depends on molar mass, pressure, and temperature. Making it a central tool for comparing gases under consistent conditions.

Molar Mass

Molar mass is an essential parameter in chemistry, referring to the mass of one mole of a substance, commonly expressed in grams per mole \((g/mol)\). It is calculated by summing the atomic masses of the constituent atoms in the molecule.
To determine the molar mass of a compound like oxygen (\(O_2\)), you would multiply the atomic mass of oxygen by two because there are two oxygen atoms. Likewise, for ammonia \((NH_3)\), add the atomic mass of nitrogen to three times the atomic mass of hydrogen.
Knowing the molar mass is crucial, as it directly influences the density of a gas. The greater the molar mass, the denser the gas will be under the same conditions of pressure and temperature. It also ties back to the Ideal Gas Law in density calculations, where molar mass is a direct factor in the formula \(\rho = \frac{Molar~Mass \times P}{RT}\). This relationship emphasizes its role in ranking gases by density.

Density Calculation

Density computation for gases involves determining how closely packed the gas particles are within a given volume. It is expressed as the mass of the gas per unit volume \((kg/m^3)\).
The density \(\rho\) can be calculated through the formula \(\rho = \frac{mass}{volume}\). For gases, we modify this formula using the Ideal Gas Law: \(\rho = \frac{Molar~Mass \times P}{RT}\), where \(P\) is the pressure and \(T\) is the temperature.

• Molar Mass: Directly impacts density; higher molar mass typically results in greater density.
• Pressure: Increasing pressure can increase density as the gas molecules are pushed closer together.
• Temperature: As temperature rises, density tends to decrease since gas molecules move apart.

This intuitive approach highlights how slight changes in conditions affect gas density and facilitates ranking gases by their densities under the same conditions, as seen in this exercise.

Periodic Table

The Periodic Table is a comprehensive chart of chemical elements arranged by atomic number, electron configuration, and recurring chemical properties. It is a fundamental tool for chemists.
Each element's square contains crucial information, including the atomic number and atomic mass, essential for calculating molar masses. This table informs the calculations of molecular compounds like \(O_2\), \(Ar\), \(NH_3\), and \(HCl\).
For example, when determining the molar mass of a compound, the atomic weight of each element (found on the Periodic Table) is used to sum up to form the compound's molar mass. The accurate determination of molar masses from the Periodic Table allows for precise density calculations.

• Facilitates chemical calculations by providing necessary atomic weights.
• Helps predict properties and behaviors of gases based on their chemical composition.

Overall, the Periodic Table serves as a starting point for any chemical calculation, ensuring accuracy and facilitating a deeper understanding of substances and their interactions.