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 van der Waals constant, \(b\), represents the volume occupied by one mole of gas particles, which depends on the molecular size. Comparing the given gases, propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) has the largest molecular size, with 11 atoms. Therefore, it has the largest value of \(b\).

Step-by-step solution

Understand the significance of the van der Waals constant, b

The van der Waals equation is an equation of state for real gases which considers the finite size of molecules and the attractive forces between them. The equation is given by: \((P + a(\frac{n^2}{V^2}))(V - nb) = nRT\) where P is the pressure, V is the volume, n is the number of moles of the gas, T is the temperature, R is the gas constant, and a and b are the van der Waals constants. The constant b represents the volume occupied by one mole of the gas particles, which depends on the size of its molecules. A larger value of b indicates larger molecules.

Compare the molecular sizes of the given gases

The molecular sizes can be estimated by comparing the number of atoms and their atomic radii (which increases down the periodic table). 1. Hydrogen (H₂): 2 atoms 2. Nitrogen (N₂): 2 atoms 3. Methane (CH₄): 5 atoms 4. Ethane (C₂H₆): 8 atoms 5. Propane (C₃H₈): 11 atoms

Determine the gas with the largest value of b

Based on the molecular sizes, propane (C₃H₈) has the largest number of atoms. Therefore, it occupies the highest volume and has the largest value for the van der Waals constant, b.

Key concepts

Van der Waals Equation

The Van der Waals equation is a modified version of the ideal gas law that accounts for the real behavior of gases. It is written as, \[\begin{equation} (P + a(\frac{n^2}{V^2}))(V - nb) = nRT \end{equation}\]where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume of the gas,
• \( n \) is the number of moles,
• \( T \) symbolizes the temperature,
• \( R \) is the ideal gas constant,
• \( a \) and \( b \) are the Van der Waals constants.

The coefficient \( a \) reflects the magnitude of the attractive forces between particles, while \( b \) takes into account the volume occupied by the gas particles themselves. This equation shows us that real gases do not always behave as predicted by the ideal gas law, especially under high pressure and low temperature conditions where the particles are closer together and their actual volume and attractions cannot be ignored.

To understand \( b \) more deeply, it represents the excluded volume – that is the volume a mole of gas molecules actually occupies and is unavailable for other molecules to occupy. Gases with larger molecules have higher values of \( b \), indicating that they need more 'personal space' and thus do not compress as much as gases with smaller molecules under the same pressure.

Real Gases Behavior

Ideal gases are a simplification for scientific calculations, as they assume no interactions between gas particles and no volume taken up by the particles. However, real gases deviate from this behavior due to two primary factors: intermolecular forces and the finite size of the gas particles. The Van der Waals equation is vital for understanding real gases because it incorporates these aspects into the calculations.

Intermolecular forces cause the gas particles to attract each other, which is particularly noticeable when a gas is compressed or cooled, making it behave differently from an ideal gas. These attractions can lead to the condensation of a gas into a liquid, which an ideal gas would never do. On the other hand, the physical volume of gas particles is addressed by the Van der Waals constant \( b \). When the gas particles are close together, their finite size matters; they cannot be squeezed indefinitely as their volume restricts how much the gas can be compressed.

The deviation from ideal behavior can be quantified using the compressibility factor, a dimensionless value that indicates how much a real gas deviates from ideal gas laws. It gives us a glimpse into the real-world applications such as in chemical engineering and thermodynamics, where the precise behavior of gases under various conditions is critical.

Molecular Size Estimation

Molecular size estimation is vital when dealing with real gases and the Van der Waals equation. The size of a molecule can influence physical and chemical properties such as boiling point, viscosity, and how the substance behaves under pressure. For the Van der Waals constant \( b \), it essentially provides a measure of the effective size of a molecule.

In the step-by-step solution, we compared the molecular sizes by considering the number of atoms within a molecule. This approach gives a rough estimate of molecular size — usually, more atoms suggest a larger molecule. However, details like structure and the nature of the chemical bonds also play a role. For instance, a long chain hydrocarbon might have a higher value of \( b \) compared to a compact, spherical-shaped molecule with the same number of atoms due to its elongated shape.

To apply this molecular size estimation to the exercise, it's clear that propane (\( C_3H_8 \)) with its larger number of atoms is expected to occupy a larger volume than the other gases listed, such as methane or hydrogen. This provides a reasonable expectation that propane would have the largest Van der Waals constant \( b \), correlating to its larger molecular size.