Question

Without looking at a table of values, which of the following gases would you expect to have the largest value of the van der Waals constant \(b : \mathrm{H}_{2}, \mathrm{N}_{2}, \mathrm{CH}_{4}, \mathrm{C}_{2} \mathrm{H}_{6},\) or \(\mathrm{C}_{3} \mathrm{H}_{8} ?\)

Short answer

The gas with the largest value of the van der Waals constant \(b\) would be \(\mathrm{C}_{3} \mathrm{H}_{8}\) (propane), as it has the largest molecular size among the given molecules.

Step-by-step solution

Understand the van der Waals constant b

The van der Waals constant b accounts for the size of the gas particles and represents the excluded volume of the gas molecules. Larger molecules will have larger values of the van der Waals constant b.

Compare the sizes of the gas molecules

Examine the given gas molecules: 1. \(\mathrm{H}_{2}\): This molecule consists of 2 hydrogen atoms. 2. \(\mathrm{N}_{2}\): This molecule consists of 2 nitrogen atoms. 3. \(\mathrm{CH}_{4}\): This molecule consists of 1 carbon atom and 4 hydrogen atoms. 4. \(\mathrm{C}_{2} \mathrm{H}_{6}\): This molecule consists of 2 carbon atoms and 6 hydrogen atoms. 5. \(\mathrm{C}_{3} \mathrm{H}_{8}\): This molecule consists of 3 carbon atoms and 8 hydrogen atoms.

Determine the gas with the largest molecular size

Comparing the number of atoms and the size of the atoms, we can rank the gas molecules from smallest to largest molecular size: 1. \(\mathrm{H}_{2}\) 2. \(\mathrm{N}_{2}\) 3. \(\mathrm{CH}_{4}\) 4. \(\mathrm{C}_{2} \mathrm{H}_{6}\) 5. \(\mathrm{C}_{3} \mathrm{H}_{8}\) Based on this ranking, the largest molecule is \(\mathrm{C}_{3} \mathrm{H}_{8}\).

Conclusion

Since the van der Waals constant \(b\) is larger for larger molecules, we can expect the gas with the largest value of \(b\) to be propane, \(\mathrm{C}_{3} \mathrm{H}_{8}\).

Key concepts

van der Waals constant b

The van der Waals constant, denoted as \(b\), is a crucial element in understanding non-ideal gas behaviors. It forms a part of the van der Waals equation, which modifies the Ideal Gas Law to account for the volume occupied by gas molecules themselves and the effects of intermolecular forces. The constant \(b\) specifically addresses the actual volume occupied by the gas molecules.

In simpler terms, \(b\) reflects the volume excluded by a mole of gas particles. Larger gas molecules tend to have larger \(b\) values because they occupy more space. As such, the van der Waals constant \(b\) provides insights into the size of gas molecules in a sample and helps in predicting their behavior under different conditions.

molecular size comparison

Understanding molecular size is key to comparing the effects of different gases using the van der Waals constant \(b\). The size of a molecule is primarily determined by the number and type of atoms it contains. A molecule with more atoms, particularly larger atoms, will naturally have a greater size.

For instance, when comparing

• \(\mathrm{H}_2\): Consists of 2 hydrogen atoms, smallest molecule here.
• \(\mathrm{N}_2\): Made up of 2 nitrogen atoms, slightly larger than hydrogen.
• \(\mathrm{CH}_4\): Contains a carbon and 4 hydrogen atoms.
• \(\mathrm{C}_2\mathrm{H}_6\): Has 2 carbon and 6 hydrogen atoms.
• \(\mathrm{C}_3\mathrm{H}_8\): Includes 3 carbon and 8 hydrogen atoms, the largest molecule in this list.

This simple comparison showcases how molecule construction directly impacts size and subsequently, the van der Waals constant \(b\). As a general rule, more atoms or heavier atoms lead to a larger molecular size.

gas molecules

Gas molecules are dynamic entities characterized by rapid, continuous motion. They have negligible volume when compared to visible scales, yet their finite size plays a significant role in real gas behavior. Understanding gas molecules involves recognizing their individual properties and how these influence gas dynamics collectively.

In an ideal gas, molecules are treated as point particles with no volume and no interaction with one another, but real gases deviate due to the actual size of molecules and their interactions.

• These molecules occupy space and exert intermolecular forces of attraction or repulsion.
• The van der Waals equation accommodates these real conditions by incorporating constants like \(a\) and \(b\).

The van der Waals constant \(b\) specifically corrects for the space occupied by gas molecules, providing a more accurate picture of gas property alterations from ideal predictions.

excluded volume

Excluded volume is a concept essential in understanding how real gases deviate from ideal behavior. In real gases, the actual volume available to a particle for motion is reduced by the presence of other particles, as they occupy space themselves.

This concept is captured within the van der Waals equation by the constant \(b\), which represents the volume excluded by a mole of particles due to their finite size. Mathematically, the excluded volume is approximately four times the actual volume of the molecules themselves.

For example:

• Larger molecules like \(\mathrm{C}_3\mathrm{H}_8\) have a greater excluded volume compared to smaller ones like \(\mathrm{H}_2\).
• The concept of excluded volume is crucial to predict the compressibility and pressure of gases under various conditions accurately.

Recognizing the importance of excluded volume helps in understanding real gas behavior and the adjustments made for deviations from the ideal gas law.