Question

Order the following gases according to increasing density: \(\mathrm{N}_{2} ; \mathrm{NH}_{3} ; \mathrm{NO}_{2}\). The temperature and pressure are the same for all three samples.

Short answer

Order of gases by increasing density: NH_{3} < N_{2} < NO_{2}.

Step-by-step solution

Write Down Molecular Masses

Calculate the molecular mass of each gas by adding up the atomic masses of the elements within the molecule. Use the periodic table for reference: Nitrogen (N) has an atomic mass of approximately 14 u, Hydrogen (H) has approximately 1 u, and Oxygen (O) has approximately 16 u.

Calculate the Molecular Masses

For N_{2}, the molecular mass is 2*14 = 28 u. For NH_{3}, the molecular mass is (1*3) + 14 = 17 u. For NO_{2}, the molecular mass is (16*2) + 14 = 46 u.

Compare the Molecular Masses

Since the temperature and pressure are constant for all three gases, their densities are directly proportional to their molecular masses.

Order by Increasing Density

The gas with the smallest molecular mass will have the lowest density, and the gas with the largest molecular mass will have the highest density. Arrange them in ascending order based on calculated molecular masses.

Key concepts

Molecular Mass

Understanding molecular mass is critical when comparing the densities of various gases. Molecular mass, sometimes referred to as molecular weight, is the sum of the atomic masses of all the atoms in a molecule. It is measured in atomic mass units (u), with each atom's mass being an average of its isotopic masses weighted by their natural abundance.

To find the molecular mass, one can use the periodic table as it lists the average atomic masses for each element. In the case of our exercise, we calculate the molecular mass of nitrogen (\textbf{N}\(_2\)) by multiplying the atomic mass of nitrogen by two. Similarly, for ammonia (\textbf{NH}\(_3\)), we sum the atomic mass of nitrogen with three times the atomic mass of hydrogen.

Ideal Gas Law

The ideal gas law is a fundamental principle that relates the pressure, volume, temperature, and amount of an ideal gas. Represented by the equation PV=nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature in Kelvins, it assumes that gases behave ideally and that gas particles do not exert forces on each other and occupy no volume.

In the context of comparing gas densities, the ideal gas law tells us that for a given pressure and temperature, the density of a gas is directly proportional to its molar mass. Even if a real gas doesn't perfectly obey the ideal gas law, the relationship serves as an excellent approximation for understanding how molecular mass affects gas density, especially under the conditions mentioned in our exercise where temperature and pressure are constant for all gases.

Molar Mass Calculation

Molar mass calculation is essentially determining the mass of one mole of a substance. The molar mass is usually expressed in grams per mole (g/mol) and is numerically equivalent to the molecular mass in atomic mass units (u).

To calculate the molar mass, we sum the masses of all atoms in the molecule. For example, Nitrogen (\textbf{N}\(_2\)) has a molecular mass of 28 u, translating to a molar mass of 28 g/mol. By calculating the molar mass of each gas, we can then compare their densities, since at constant temperature and pressure, the gas with higher molar mass will be denser. As evident in our exercise, understanding how to compute the molar mass is indispensable for ordering gases by their density.