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

Based on the molecular structure and size, we expect that the gas with the largest value of the van der Waals constant \(b\) would be \(\mathrm{C}_{3} \mathrm{H}_{8}\).

Step-by-step solution

Understanding the molecular structure

With each molecule, the number of carbon (C) and hydrogen (H) atoms will determine the molecular size. More carbon and/or hydrogen atoms in the molecule increase the molecule's size.

Comparing the given molecules

Now, let's compare the number of carbon and hydrogen atoms for each molecule: 1. \(\mathrm{H}_{2}\): 0 carbon atoms, 2 hydrogen atoms 2. \(\mathrm{N}_{2}\): 0 carbon atoms, 0 hydrogen atoms (nitrogen gas) 3. \(\mathrm{CH}_{4}\): 1 carbon atom, 4 hydrogen atoms 4. \(\mathrm{C}_{2}\mathrm{H}_{6}\): 2 carbon atoms, 6 hydrogen atoms 5. \(\mathrm{C}_{3} \mathrm{H}_{8}\): 3 carbon atoms, 8 hydrogen atoms

Identifying the gas with the largest molecular size

Now that we have compared the number of carbon and hydrogen atoms for each molecule, we can see that \(\mathrm{C}_{3} \mathrm{H}_{8}\) has the most carbon (3) and hydrogen (8) atoms.

Conclusion

Based on the molecular structure and size, we expect that the gas with the largest value of the van der Waals constant \(b\) would be \(\mathrm{C}_{3} \mathrm{H}_{8}\).

Key concepts

Molecular Structure

Molecular structure refers to the arrangement of atoms within a molecule. It determines a molecule's properties and how it interacts with other substances. In the context of van der Waals forces, which are weak attractions between molecules, the structure is crucial. These forces become more significant as molecules become larger and more complex in structure. This complexity often results in a larger molecular "surface area" for interaction. Understanding molecular structure helps predict how molecules will behave in various physical conditions, such as changes in temperature or pressure. For example, more complex structures can lead to higher boiling points. This is because it takes more energy to break the intermolecular forces in a large, complex molecule compared to a smaller, simpler one. In the original exercise, recognizing the differences in molecular structure helps predict which gas has the largest van der Waals constant, an indicator of molecular size and interaction capabilities.

Carbon and Hydrogen Atoms

Atoms are the building blocks of molecules, and both carbon and hydrogen play pivotal roles in organic chemistry. Carbon atoms can form four bonds due to their four valence electrons, allowing for a variety of configurations. Hydrogen atoms typically make one bond. Together, these atoms create diverse organic compounds, including hydrocarbons like methane (\(\mathrm{CH}_4\)), ethane (\(\mathrm{C}_2\mathrm{H}_6\)), and propane (\(\mathrm{C}_3\mathrm{H}_8\)).

• Carbon: Tends to form long chains or rings, contributing significantly to a molecule's size and complexity.
• Hydrogen: Often controls the saturation of a molecule, influencing its reactivity and boiling point.

In the exercise, by counting the carbon and hydrogen atoms, we determine the relative sizes of different molecules. More carbon and hydrogen atoms generally equate to a larger molecule. This has direct implications for the van der Waals constant, as larger molecules tend to have higher b values due to the increased area available for intermolecular interactions.

Molecular Size

Molecular size fundamentally affects a molecule's physical and chemical properties. Larger molecules exhibit stronger van der Waals forces due to increased surface areas available for interactions. This is why molecular size is a crucial consideration in the context of the van der Waals constant (denoted \(b\)). The van der Waals constant \(b\) is part of a modified ideal gas law, accounting for the volume occupied by gas particles themselves. Larger molecules have larger b values because they deplete more space. For instance, in our exercise with gases like hydrogen (\(\mathrm{H}_2\)) and propane (\(\mathrm{C}_3\mathrm{H}_8\)), propane's larger size results in a higher b constant. This indicates both a physically larger size and a greater propensity for intermolecular attraction; facts directly derived from comparing molecular sizes. Recognizing the connection between these properties is essential for understanding gas behavior under various conditions, providing insight into why molecular size matters in practical applications such as chemical reactions and industrial processes.