Van Der Waals Equation

Van der Waals equation is an equation of state that describes the behavior of real gases, taking into account the finite size of gas molecules and the attractive forces between them. Van Der Waals equation is written like this: [P + a (n²/V²)] (V-nb) = nRT. The Van der Waals equation stands as a crucial development in understanding the behavior of real gases. It is very similar to the Ideal Gas Law PV = nRT except that for attraction between gas molecules with ‘a’ and volume of those molecules with ‘b’.

In this article, we will learn about, Van Der Waals Equation, Van Der Waals Equation Formula, Van Der Waals constants, Van Der Waals Equation Derivation, and others in detail.

What is Van Der Waals Equation?

Van der Waals equation, proposed by Dutch physicist Johannes Diderik van der Waals in 1873, addresses the limitations of the ideal gas law.

Ideal Gas Law

Ideal gas law states that,

PV = n × R × T

where,

• P is Pressure
• V is Volume
• n is Number of Moles
• R is Gas Constant
• T is Temperature

Assuming that gas molecules have no volume and do not exert attractive forces on each other. However, real gases deviate from this ideal behavior, especially at high pressures and low temperatures.

Van Der Waals Equation Formula

Van Der Waals equation adjusts ideal gas law to account for volume occupied by gas molecules and attractive forces between them. The equation is given by:

[P + a (n²/V²)] (V – nb) = nRT

where,

• P is Pressure
• V is Volume
• n is Number of Moles
• T is Temperature
• a is Correction Factor for Attractive Forces Between Molecules
• b is Correction Factor for Volume Occupied by Gas Molecules

Term a(n²/V²) corrects for Attractive Forces, and nb corrects for Volume Occupied by Gas Molecules. Van Der Waals equation is particularly useful in describing behavior of real gases under conditions where deviations from ideal behavior are significant.

Van Der Waals Constants

Van Der Waal’s constant are,

• ‘a’ Parameter: Reflects the attractive forces between gas molecules. Higher values of ‘a’ indicate stronger intermolecular forces.
• ‘b’ Parameter: Accounts for the volume occupied by gas molecules. It represents the correction for the finite size of gas particles.

Units of Van Der Waals Constant

• a: Measured in (Pa⋅m⁶)/mol².
• b: Measured in m³/mol.

Need for Van Der Waals Equation

Real gases deviate from ideal behavior under conditions where the assumptions of the ideal gas law break down. This occurs notably at high pressures and low temperatures. In these scenarios, intermolecular forces and the finite volume of gas particles become significant. Unlike ideal gases, real gases experience attractive and repulsive forces between molecules, leading to deviations in their behavior. The Van der Waals equation accounts for these deviations by introducing correction terms (‘a’ for attractive forces and ‘b’ for volume occupied) and provides a more accurate representation of real gas behavior under diverse conditions.

Van Waal Theory of Gas

Van der Waals equation, proposed by Johannes Diderik Van Der Waals in 1873, addresses the limitations of the ideal gas law. This theory acknowledges that real gas molecules occupy space and experience attractive forces between them, aspects overlooked in the ideal gas model.
The equation introduces correction terms a and b to account for molecular interactions and the finite volume occupied by gas particles. a represents the strength of attractive forces, while b adjusts for volume occupied by gas molecules.
Van der Waals theory provides a more accurate depiction of gas behavior under non-ideal conditions, contributing to a deeper comprehension of the intricacies of gas interactions.

Van Der Waals Equation Derivation

Using ideal gas equation,

PV = nRT

This equation assumes that gas molecules have negligible volume and do not experience attractive or repulsive forces.

However, real gases deviate from these idealized conditions, particularly at high pressures and low temperatures.

Correcting for Volume

First correction factor, nb, is introduced to account for the volume occupied by gas molecules.

Corrected volume becomes (V – nb). Thus, final equation will be:

P(V – nb) = nRT

Correcting for Attractive Forces

To correct for attractive forces between gas molecules, a term proportional to is added. This term reflects the reduction in pressure due to attractive forces, making the pressure term in the equation

((P + a(n²)/V²)⋅(V) = nRT

Here, a is a constant that represents the strength of attractive forces between molecules. Term a(n²)/V² can be seen as an adjustment to the pressure term to account for the reduction caused by attractive forces.

Combining Corrections

Combining these corrections with the ideal gas law yields the Van der Waals equation:

((P + a(n²)/V²) ⋅ (V-nb) = nRT

Equation is more accurate for describing the behavior of real gases, especially under conditions where deviations from ideal behavior are significant.

Van Der Waals equation is a crucial advancement over Ideal Gas Law, providing a more realistic representation of gas behavior by considering both molecular volume and intermolecular forces.

Advantages and Disadvantages of Van Der Waals Equation

There are several advantages and disadvantages of Van Der Waals equations some of these are listed below:

Advantages of Van Der Waals Equation

• Realistic Representation: Corrects ideal gas law assumptions.
• Precision: Applicable under various conditions.
• Industrial Applicability: Crucial in industries like chemical manufacturing.

Disadvantages of Van Der Waals Equation

• Limited Accuracy: Faces challenges in extreme conditions.
• Complex Molecular Interactions: May struggle with highly complex molecular scenarios.
• Phase Transition Prediction: Difficulty in accurately predicting phase transitions.

Applications of Van der Waals Equation

1. Drug Design and Pharmaceutical Industry: Predicting drug solubility and optimizing drug-receptor interactions.
2. Materials Science and Engineering: Employed to forecast and control material properties.
3. Environmental Chemistry: Assists in studying gas behavior in the atmosphere for pollution control.
4. Chemical Engineering Processes: Ensures precision in reactions under various conditions.
5. Biological Systems and Biomolecular Interactions: Contributes to understanding biomolecular interactions in life processes.
6. Astrophysics and Gas Behavior in Space: Predicts gas behavior in space, aiding our understanding of celestial bodies.

Ideal Gas Equation vs Van der Waals Equation: A Comparative Analysis

Ideal Gas Equation

Ideal gas equation,

PV = nRT

It provides a simplified model for the behavior of gases by assuming that gas molecules have negligible volume and do not exert attractive or repulsive forces on each other. While it serves as a useful approximation under many conditions, real gases deviate from ideal behavior, especially at high pressures and low temperatures.

Van der Waals Equation

Van der Waals equation, proposed by Johannes Diderik van der Waals in 1873, offers a more comprehensive description of real gas behavior. It introduces corrections to the ideal gas equation to account for the finite size of gas molecules and the attractive forces between them. The equation is given by:

((P + a(n²)/V²)⋅(V – nb) = nRT

Here, a corrects for attractive forces, and b corrects for the volume occupied by gas molecules.

Property	Ideal Gas Equation	Van der Waals Equation
Molecular Size	Assumes point-like particles with no volume.	Accounts for the finite volume of gas molecules (term b in the equation).
Intermolecular Forces	Neglects intermolecular forces.	Includes correction for attractive forces between molecules (term a in the equation).
Pressure Correction	No correction for high pressures.	Corrects for high pressures by adding a⋅(n2/V2)to the pressure term.
Volume Correction	No correction for the volume occupied by gas.	Adjusts the volume term by subtracting nb from V to account for the excluded volume of gas molecules.
Applicability	Suitable for low pressures and high temperatures.	More accurate at high pressures and low temperatures where molecular volumes and attractive forces become significant.
General Form	PV = nRT	[P + a (n2/V2)] (V-nb)

Contrasting the ideal gas law with the Van der Waals equation highlights the limitations of the former and the advancements provided by the latter. The ideal gas law assumes negligible volume of gas particles and lacks considerations for intermolecular forces. In situations where these assumptions are invalid, such as high-pressure conditions, the ideal gas law falls short, while the Van der Waals equation provides a more accurate description. The Van der Waals equation becomes essential in addressing the shortcomings of the ideal gas law, offering a refined model for real gas behavior.

Experimental Verification and Validation

Historical and contemporary experiments validate predictions of Van Der Waals equation. Notable experiments, such as those conducted by van der Waals himself and subsequent researchers, confirm the accuracy of the equation in describing real gas properties. These experiments involve varying conditions of temperature and pressure to test the equation’s applicability across a range of scenarios.

Limitations of Van Der Waals Equation

Struggles with Extreme Conditions: Van der Waals equation faces challenges when attempting to model gas behavior under extreme conditions of temperature or pressure.

Complex Molecular Interactions: Its accuracy diminishes in cases of highly complex molecular interactions, limiting its effectiveness in certain scenarios.

Inability to Capture Phase Transitions: Equation may struggle to accurately predict phase transitions, especially in situations where rapid changes in state occur.

Sensitivity to Molecular Structure: Sensitivity to molecular structure variations can impact the equation’s precision, making it less reliable for certain molecules or chemical configurations.

Read, More

• Gay Lussac’s Law Formula
• Newton’s Law of Cooling
• Charles Law

Examples on Van Der Waals Equation

Example 1: Calculate the Pressure for a gas of Volume 1 m³, with the following conditions: n = 2 moles, T = 300K, a = 3.5Pa⋅m⁶/mol², b = 0.042m³/mol.

Solution:

Substitute given values into Van Der Waals Equation: ((P + a(n²)/V²)⋅(V − nb) = nRT

((P + 3.5(2²)/1²) × (1 − 2⋅0.042) = 2×8.314×300

(P + 14/1)(1 – 0.084) = 4988.4

P + 14 = 5445.85

P = 5431.85 Pa

P ≈ 5.431 × 10³Pa

Example 2: A gas sample has a volume of 2.00 L, a pressure of 3.00 atm, and a temperature of 300 K. For this gas, n=2 moles and b=0.030L/mol. Find value of a.

Solution:

Given,

• P = 3.00atm
• V = 2.00L
• T = 300K
• n = 2
• b = 0.030L/mol
• R (gas constant) = 0.0821 L⋅atm/mol.K

Substitute given values into Van Der Waals Equation:

((P + a(n²)/V²)×(V − nb) = nRT

(3 + a(2²)/ (2²))⋅(2-2⋅0.03) =2×0.0821×300

(3+a⋅1) ⋅ (1.94) =49.26

3 + a = 25.39

a = 22.39

a = 22.39 L² atm/mol²

Example 3: A gas sample has a volume of 1.00 L, a pressure of 10.00 atm, and a temperature of 300 K. For this gas, n= 5 moles and a=5 L² atm/mol. Find the value of b.

Solution:

Given,

• P = 10.00atm
• V = 1.00L
• T = 300K
• n = 5 moles
• a = 5L² atm/mol
• R (Gas Constant) = 0.0821 L⋅atm/mol⋅K

Substitute given values into Van Der Waals equation: ((P + a(n²)/V²)⋅(V − nb) = nRT

(10 + 5(5⁵/1))⋅(1 – 5⋅b) = 1×0.0821×300

1 – 5⋅b=24.63/135

-5⋅b = 1.82 – 1

b = (-0.82)/(-5)

b = 0.163

b ≈ 0.16 L/mol

Example 4: Calculate the Pressure for a gas of Volume 1 m³, with the following conditions: n = 2 moles, T = 300K, a=8 Pa⋅m⁶/mol², b=0.02 m³/mol.

Solution:

Given,

• V = 1 m³
• T = 300K
• n = 2 moles
• a = 8 Pa⋅m⁶/mol²
• b = 0.02 m³/mol
• R (Gas Constant) = 8.314 J/mol⋅K

Substitute given values into Van Der Waals Equation,

((P + a(n²)/V² )⋅(V − nb) = nRT

((P + 8(2²) / 1² )⋅(1 − 2 × 0.02) = 2 × 8 × 314 × 300

(P + 32) (1 – 0.04) = 4988.4

P + 32 = (4988.4)/(0.96)

P = 5196.25 – 32

P = 5164.25

P = 5.164 × 10³ Pa

Problems on Van Der Waals Equation

Various problems on Van Der Waals Equations are,

P1: Calculate Volume for a gas at Pressure 10³ Pa, with the following conditions: n = 3 moles, T = 400K, a=2 Pa⋅m⁶/mol², b=0.002m³/mol.

P2: Determine value of correction factor constant a with Volume 1 m³ for a gas at a Pressure 10³ with the following conditions: n= 2 moles, T = 300K, b = 0.042 m³/mol.

P3: Determine value of correction factor constant a with Volume 18 m³ for a gas at a Pressure 16³ with the following conditions: n= 3 moles, T = 300K, b = 0.042 m³/mol.

FAQs on Van Der Waals Equation

What is Van Der Waals Equation, and How Does It Differ from Ideal Gas Law?

Van der Waals equation is a modification of the ideal gas law, accounting for finite molecular size and intermolecular forces absent in the ideal gas law.

What Do Constants a and b Mean in Van Der Waals Equation?

• a: Represents strength of attractive forces between gas molecules. Higher a indicates stronger intermolecular attractions.
• b: Volume correction term that adjusts for the finite size of gas particles.

What is Van Der Waals Theory of Gases?

Van der Waals theory extends the ideal gas model by considering finite particle size and intermolecular attractions. The resulting Van der Waals equation provides a more accurate description of real gas behavior, especially at high pressures and low temperatures.

What are Assumptions of Van Der Waals Theory of Gases?

• Gas particles have finite volumes and occupy space.
• Attractive forces exist, proportional to n².
• Continuous random motion of gas particles.
• Negligible volume of gas particles compared to the overall gas volume.

What is Critical Point of a Van der Waals Gas?

The critical point is the set of conditions (critical temperature, pressure, and volume) where a Van der Waals gas undergoes a phase transition from gas to liquid without distinction between the two phases. At the critical point, the gas and liquid phases become indistinguishable, marking a unique behavior of real gases under extreme conditions.

Can Van der Waals Equation Accurately Predict Gas Behavior Under Extreme Conditions?

While, Van Der Waals equation is an improvement over ideal gas law, it may struggle under extreme conditions, such as very high pressures or temperatures.

Are there Limitations to Van Der Waals Equation in Capturing Complex Molecular Interactions?

Yes, Van der Waals equation may face challenges in accurately capturing highly complex molecular interactions, leading to limitations in certain scenarios.

Why is Van der Waals Equation Crucial in Understanding Real Gas Behavior?

It is crucial because it corrects for the ideal gas law’s limitations, considering molecular size and intermolecular forces, providing a more accurate representation of real gas behavior.