Problem 48 - Chemistry The Central Science In Si Units

Question

Rank the following gases and vapors from least dense to most dense at \(101.33 \mathrm{kPa}\) and \(298 \mathrm{~K}:\) water vapor \(\left(\mathrm{H}_{2} \mathrm{O}(g)\right),\) nitrogen \(\left(\mathrm{N}_{2}\right),\) hydrogen sulfide \(\left(\mathrm{H}_{2} \mathrm{~S}\right)\)

Step-by-Step Solution

Answer

The order from least dense to most dense is water vapor, nitrogen, hydrogen sulfide.

Step 1: Identify the Molar Mass of Each Gas

Calculate the molar mass of each gas or vapor using the periodic table: \( \text{H}_2\text{O}(g) \) has a molar mass of approximately \( 18.02 \text{ g/mol} \) (2 Hydrogen atoms at \( 1.01 \text{ g/mol each} \) + 1 Oxygen atom at \( 16.00 \text{ g/mol} \)), \( \text{N}_2 \) has a molar mass of \( 28.02 \text{ g/mol} \) (2 Nitrogen atoms at \( 14.01 \text{ g/mol each} \)), and \( \text{H}_2\text{S} \) has a molar mass of approximately \( 34.08 \text{ g/mol} \) (2 Hydrogen atoms at \( 1.01 \text{ g/mol each} \) + 1 Sulfur atom at \( 32.06 \text{ g/mol} \)).

Step 2: Arrange the Gases by Molar Mass

Order the gases by increasing molar mass: \( \text{H}_2\text{O}(g) \) has the lowest molar mass (18.02 g/mol), followed by \( \text{N}_2 \) (28.02 g/mol), and finally \( \text{H}_2\text{S} \) has the highest molar mass (34.08 g/mol).

Step 3: Identify Density as a Function of Molar Mass

Since all gases are at the same conditions of temperature and pressure, their densities are directly proportional to their molar masses. Thus, the ordering of densities will be the same as the ordering of molar masses.

Step 4: Conclude the Density Order

From the previous steps, the order from least dense to most dense is: water vapor \((\text{H}_2\text{O}(g))\), nitrogen \((\text{N}_2)\), and hydrogen sulfide \((\text{H}_2\text{S})\).

Key Concepts

Molar Mass

Molar mass is a key concept in understanding gas density. It is the mass of one mole of a given substance, usually expressed in grams per mole (g/mol). To find the molar mass of a compound, you add the masses of all the atoms in a molecule, using their respective values from the periodic table.

For example, let's examine water vapor \(\text{H}_2\text{O(g)}\). This molecule comprises two hydrogen atoms, each with a molar mass of approximately 1.01 g/mol, and one oxygen atom with a molar mass of 16.00 g/mol. When you add these together, the molar mass of water vapor is about 18.02 g/mol.

Similarly, nitrogen \(\text{N}_2\) contains two nitrogen atoms, each with a molar mass of 14.01 g/mol, giving it a total molar mass of approximately 28.02 g/mol. Hydrogen sulfide \(\text{H}_2\text{S}\) consists of two hydrogen atoms at 1.01 g/mol each and one sulfur atom at 32.06 g/mol, resulting in a molar mass of around 34.08 g/mol. Understanding molar mass helps us directly determine the density and behavior of gases under standard conditions.

Gas Properties

Gases possess unique properties affecting their behavior, such as density, which depends on factors including molar mass, temperature, and pressure.

At constant temperature and pressure, as in our exercise, the density of a gas is directly proportional to its molar mass. This means that gases with a higher molar mass will be more dense than those with a lower molar mass. This is because density is determined by how much mass occupies a certain volume: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

Another important property of gases is diffusion, the movement from areas of higher concentration to lower concentration. Gases with lower molar mass, like water vapor, tend to diffuse more rapidly than those with higher molar mass, like hydrogen sulfide. These properties play a significant role in chemical reactions and physical processes involving gases.

Periodic Table Usage

The periodic table is an indispensable tool for chemists, providing critical information needed to calculate molar mass and understand elemental properties.

To determine the molar mass of any compound, you begin by identifying the constituent elements in a given molecule. For each element, the periodic table provides the atomic mass, typically found below the element symbol. By multiplying this atomic mass by the number of each type of atom in the molecule, you'll be able to compute its total molar mass.

For instance, the atomic mass of hydrogen is approximately 1.01 g/mol and can be found on the periodic table. Similarly, the atomic masses for elements like oxygen (16.00 g/mol), nitrogen (14.01 g/mol), and sulfur (32.06 g/mol) are all readily accessible on this table. By employing these values, we can solve problems involving gaseous substances systematically and accurately. The periodic table remains crucial not only for the calculation of molar mass but also for predictions related to chemical reactions and the properties of elements.