Van der Waals Equation: Videos & Practice Problems

The Van der Waals equation refines the Ideal Gas Law to account for real gas behavior, which deviates from the ideal at high pressures and low temperatures. This equation introduces two constants: 'a' for attractive forces (polarity) between gas molecules and 'b' for the volume occupied by the gas molecules (size). The 'b' constant increases with the molecular weight of a gas, reflecting its larger volume. The Van der Waals equation serves as a bridge between real and ideal gases, reverting to the Ideal Gas Law when the volume and attractive forces of the gas molecules are negligible. Understanding the implications of the 'a' and 'b' constants is more crucial than memorizing the equation, as it's typically provided when needed for calculations.

Van der Waals Equation

The van der Waals equation is essential for understanding the behavior of real gases, particularly under conditions where they deviate from ideal gas behavior, such as at high pressures and low temperatures. Unlike ideal gases, which are theoretical constructs that assume gas molecules do not interact and occupy no volume, real gases experience significant interactions and volume constraints. At high pressures, gas molecules are forced closer together, disrupting the ideal assumption of isolation. Similarly, low temperatures can lead to condensation, further increasing the proximity of gas molecules.

The van der Waals equation incorporates two important constants, denoted as a and b. The constant a accounts for the attractive forces between gas molecules, while b corrects for the finite volume occupied by the gas molecules themselves. It is noteworthy that as the molecular weight of a gas increases, the value of the van der Waals constant b also increases, indicating a direct relationship between molecular weight and the volume correction factor.

In summary, the van der Waals equation provides a more accurate representation of real gas behavior by incorporating these corrections, allowing for better predictions and understanding of gas properties under non-ideal conditions.

Van de Waals Theory

The Van der Waals equation is essential for understanding the behavior of real gases, as it accounts for the interactions between gas molecules, which the ideal gas law does not. The equation is expressed as:

\[P + \frac{n^2 a}{V^2} = \frac{nRT}{V - nb}\]

In this equation, \(P\) represents pressure, \(n\) is the number of moles, \(R\) is the ideal gas constant, \(T\) is temperature, \(V\) is volume, and the constants \(a\) and \(b\) are specific to each gas. The term \(\frac{n^2 a}{V^2}\) corrects for the attractive forces between gas molecules, while the term \(nb\) accounts for the volume occupied by the gas molecules themselves.

Understanding the significance of the constants \(a\) and \(b\) is crucial. The constant \(a\) reflects the strength of the attractive forces between molecules; higher values indicate stronger attractions. The constant \(b\) represents the volume occupied by one mole of the gas, which correlates with the size of the gas molecules. Generally, as the molecular weight of a gas increases, the value of \(b\) also tends to increase, although there may be some exceptions due to the complexities of chemical behavior.

It's important to note that the Van der Waals equation simplifies to the ideal gas law under certain conditions. For an ideal gas, the volume of the gas molecules is negligible, leading to \(b = 0\), and the attractive forces are absent, resulting in \(a = 0\). Thus, the Van der Waals equation reduces to:

\[PV = nRT\]

This connection highlights the limitations of the ideal gas law, which assumes perfect conditions that do not exist in reality. Therefore, while the ideal gas law is a useful approximation, the Van der Waals equation provides a more accurate representation of gas behavior under real-world conditions.

Van der Waals Pressure Example

The van der Waals equation is a modification of the ideal gas law that accounts for the volume occupied by gas molecules and the attractive forces between them. The equation is expressed as:

\( P + \frac{2a}{V^2} \cdot V - n \cdot b = nRT \)

In this scenario, we need to determine the pressure of 20 grams of oxygen gas (\(O_2\)) in a 150 mL graduated flask at a temperature of 50 degrees Celsius. First, we convert the mass of oxygen to moles. Since oxygen is diatomic, one mole of \(O_2\) weighs 32 grams. Thus, the number of moles (\(n\)) can be calculated as:

\( n = \frac{20 \text{ g}}{32 \text{ g/mol}} = 0.625 \text{ moles} \)

Next, we convert the volume from milliliters to liters:

\( V = 150 \text{ mL} = 0.150 \text{ L} \)

We also need to convert the temperature from Celsius to Kelvin:

\( T = 50 + 273.15 = 323.15 \text{ K} \)

For the van der Waals constants for oxygen, we use \(a = 1.360 \, \text{L}^2 \cdot \text{atm/mol}^2\) and \(b = 0.0318 \, \text{L/mol}\).

Now, we can substitute these values into the van der Waals equation. Rearranging the equation to isolate \(P\), we have:

\( P = nRT - \frac{2a}{V^2} \cdot V + n \cdot b \)

Substituting the known values:

\( P = (0.625 \, \text{mol}) \cdot (0.08206 \, \text{L} \cdot \text{atm/(mol} \cdot \text{K)}) \cdot (323.15 \, \text{K}) - \frac{2 \cdot 1.360}{(0.150)^2} \cdot (0.150) + (0.625 \cdot 0.0318) \)

Calculating the right side:

\( P = 16.57356 - \frac{2.720}{0.0225} + 0.019875 \)

Calculating the pressure contributions:

\( P = 16.57356 - 120.88889 + 0.019875 \)

After simplifying, we find:

\( P + 8.5 = 72.00 \)

Finally, isolating \(P\) gives:

\( P = 63.5 \, \text{atm} \)

Thus, the pressure of the oxygen gas in the flask, using the van der Waals equation, is 63.5 atmospheres.

Which gaseous compound is expected to have the largest value for the Van der Waals constant b?

• A. O₂
• B. CH₃CH₃
• C. HBr
• D. S₈
• E. Ne

Answer: (D)

Here’s what students ask on this topic:

What is the Van der Waals equation?

The Van der Waals equation is a mathematical formula used to describe the behavior of real gases, taking into account the volume occupied by gas molecules and the attractive forces between them, which are not considered in the ideal gas law. It is an improvement over the ideal gas law and is expressed as:

( P + a V 2 ) ( V m - b ) = R T Where:

• P is the pressure of the gas,
• V_m is the molar volume of the gas (the volume per mole of gas),
• T is the temperature of the gas in Kelvin,
• R is the ideal gas constant,
• a is a measure of the attraction between the particles,
• b is the volume occupied by one mole of the particles themselves.

The constants a and b are specific to each gas, reflecting how different gases have different sized molecules and strengths of intermolecular forces. The Van der Waals equation corrects for the non-ideal behavior of gases, especially at high pressures and low temperatures, where deviations from the ideal gas law are most pronounced.

How can I find the values of a and b in the van der Waals equation?

To find the values of a and b in the van der Waals equation for a specific gas, you typically refer to empirical data. The van der Waals equation is an equation of state for a fluid composed of particles with non-zero size and a pairwise attractive force, represented as:

( P + a n 2 V 2 ) ( V - n b ) = n R T

Here, P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. The constants a and b are unique to each gas and are determined experimentally. They account for the intermolecular forces and the volume occupied by the gas particles, respectively.

To find these values, you would look them up in physical chemistry textbooks, scientific databases, or reputable online resources. These constants are typically derived from experimental measurements of the gas properties at various temperatures and pressures and then fitted to the van der Waals equation. Once known, a and b can be used to predict the behavior of the gas under non-ideal conditions.

What does the van der Waals equation account for?

The van der Waals equation is a mathematical expression that improves upon the ideal gas law by accounting for two significant factors that affect real gases: intermolecular forces and the finite size of gas particles.

Firstly, it introduces a term to correct for the intermolecular attractions between gas molecules. In an ideal gas, these forces are assumed to be negligible, but in real gases, they can have a noticeable impact, especially under high pressure. The van der Waals equation adds an attraction term that effectively reduces the pressure in the ideal gas equation to account for these forces.

Secondly, it corrects for the volume occupied by the gas particles themselves. The ideal gas law assumes that the particles have no volume, but real gas particles do occupy space. The van der Waals equation adds a volume correction term to the ideal gas law, which effectively increases the volume over which the gas particles are free to move.

By including these two corrections, the van der Waals equation provides a more accurate description of the behavior of real gases under various conditions than the ideal gas law.

What advantage does the van der Waals equation give us over the ideal gas law?

The van der Waals equation provides a more accurate description of the behavior of real gases compared to the ideal gas law. While the ideal gas law assumes that gas particles have no volume and do not exert any attractive forces on each other, these assumptions are not true for real gases. The van der Waals equation modifies the ideal gas law by introducing two constants, a and b. The a term accounts for the attractive forces between gas molecules, which become significant at high pressures and low temperatures. The b term corrects for the finite volume that gas molecules occupy, which becomes important at high pressures where the volume of the gas particles is no longer negligible compared to the volume of the container. By including these factors, the van der Waals equation can more accurately predict the behavior of gases under non-ideal conditions, such as high pressures and low temperatures, where deviations from ideal behavior are most pronounced.