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 C3H8 (Propane), as it has the largest molecular weight (\(44.096 \, amu\)) among the given gases. This is because molecular weight plays a significant role in determining the van der Waals constant \(b\).

Step-by-step solution

Analyzing the molecular weight

To determine which gas would have the largest value of the van der Waals constant b, we'll need to consider the molecular weight of each gas. Remember that the molecular weight is determined by the sum of the atomic weights of the atoms in the molecule.

Evaluating each gas's molecular weight

Let's evaluate the molecular weight of each gas: 1. H2 (Hydrogen gas): Its molecular weight is 2(1.008) = \(2.016 \, amu\) 2. N2 (Nitrogen gas): Its molecular weight is 2(14.007) = \(28.014 \, amu\) 3. CH4 (Methane): Its molecular weight is 12.011 + 4(1.008) = \(16.043 \, amu\) 4. C2H6 (Ethane): Its molecular weight is 2(12.011) + 6(1.008) = \(30.069 \, amu\) 5. C3H8 (Propane): Its molecular weight is 3(12.011) + 8(1.008) = \(44.096 \, amu\)

Identifying the gas with the largest molecular weight

Comparing the molecular weight of each gas, we can see that C3H8 (Propane) has the largest molecular weight of \(44.096 \, amu\).

Determining the gas with the largest value of the van der Waals constant b

Since molecular weight plays a significant role in determining the van der Waals constant b, we can conclude that C3H8 (Propane) would have the largest value of the van der Waals constant b among the given gases due to its largest molecular weight.

Key concepts

Molecular Weight

Molecular weight is a critical factor when assessing the properties of a gas, as seen in our problem examining the van der Waals constant. Molecular weight is calculated by summing up the atomic weights of the atoms present in a molecule. For each gas:

• Hydrogen gas, \( \mathrm{H}_2 \), with a molecular weight of approximately \(2.016 \, \text{amu} \).
• Nitrogen gas, \( \mathrm{N}_2 \), weighs in at \(28.014 \, \text{amu} \).
• Methane, \( \mathrm{CH}_4 \), has a molecular weight of \(16.043 \, \text{amu} \).
• Ethane, \( \mathrm{C}_2\mathrm{H}_6 \), comes in at \(30.069 \, \text{amu} \).
• Propane, \( \mathrm{C}_3\mathrm{H}_8 \), possesses the largest molecular weight at \(44.096 \, \text{amu} \).

These weights are rounded based on the atomic masses of hydrogen, carbon, and nitrogen. Because molecular weight impacts volume and interaction characteristics of gases, it is a foundational metric in predicting gas behavior, including influences on the van der Waals constants.

Gas Properties

Gas properties are vital in understanding how gases behave under different conditions. Gases are composed of many molecules that move freely and rapidly. The behaviors of gases can be influenced by several key properties:

• Volume: Gases occupy space and the volume affects the space that gases fill.
• Pressure: Gas molecules collide with container walls, creating pressure.
• Temperature: Increasing temperature raises the kinetic energy of gas molecules.
• Molecular Interactions: At higher pressures or lower temperatures, gases exhibit non-ideal behavior due to intermolecular forces.
• Molecular Size: Larger molecules generally occupy more space and experience stronger interactions.

Understanding these properties helps evaluate equations of states, like the ideal gas law and van der Waals equation, predicting gas behavior under various circumstances.

van der Waals Equation

The van der Waals equation is a mathematical formula used to describe the behavior of real gases, providing a way to account for deviations from the ideal gas law. The equation incorporates two critical correction factors:\[\left( P + \frac{a}{V_m^2} \right)(V_m-b) = RT\]Where:

• \( P \) stands for pressure of the gas.
• \( V_m \) represents the molar volume.
• \( a \) corrects for intermolecular forces.
• \( b \) corrects for the finite size of molecules (related to molecular volume).
• \( R \) is the ideal gas constant.
• \( T \) is the temperature.

The constant \( b \) particularly reflects the volume occupied by gas molecules. Larger molecules with higher molecular weights tend to have larger values of \( b \), as seen in the exercise where propane has the largest \( b \). This means they take up more space in the gas phase. The van der Waals equation helps predict how real gases deviate from ideal behavior by accounting for the particle volume and intermolecular forces.