comparative table of differences between the van der waals, redlich-kwong and peng-robinson equations
The Correct Answer and Explanation is :
Here’s a comparative table of the Van der Waals, Redlich-Kwong, and Peng-Robinson equations of state:
| Feature | Van der Waals | Redlich-Kwong | Peng-Robinson |
|---|---|---|---|
| Equation Form | [(P + \frac{a}{V^2})(V – b) = RT] | [P = \frac{RT}{V-b} – \frac{a}{\sqrt{T}V(V+b)}] | [P = \frac{RT}{V-b} – \frac{a(T)}{V(V+b) + b(V-b)}] |
| Attraction Term (a) | (\frac{a}{V^2}) | (\frac{a}{\sqrt{T}V(V+b)}) | (a(T)) function of temperature and reduced properties |
| Repulsion Term (b) | ((V – b)) | ((V – b)) | ((V – b)) |
| Temperature Dependence | None (constant a, b) | Dependent on T (a is a function of T) | Temperature dependence through a(T) |
| Applicability | Good for gases near ideal behavior | Better than Van der Waals for low pressures | Better accuracy for gases at higher pressures and near the critical point |
| Accuracy | Moderate | Better than Van der Waals at moderate T and P | Most accurate for a wide range of pressures and temperatures |
| Critical Point Behavior | Cannot accurately predict the critical point | Improved but still limited | More accurate prediction of critical properties |
| Compressibility Factor | Overestimates for non-ideal gases | Better than Van der Waals | Closest to experimental values |
| Primary Use | Historical importance; simple calculations | Used for moderate accuracy in engineering | Widely used for modern applications, especially in petroleum and natural gas industries |
Explanation
The Van der Waals, Redlich-Kwong, and Peng-Robinson equations are different models used to describe the behavior of real gases, particularly their deviations from ideal gas behavior.
- Van der Waals Equation: One of the earliest attempts to modify the ideal gas law to account for molecular volume and intermolecular forces. It introduces the constants (a) (attractive forces) and (b) (molecular volume). This equation works reasonably well for gases at low pressures but is not accurate near the critical point or at high pressures.
- Redlich-Kwong Equation: An improvement over Van der Waals, it introduces a temperature dependence to the attractive term, improving accuracy for gases at higher pressures and moderate temperatures. It is often used in chemical engineering applications where moderate precision is needed.
- Peng-Robinson Equation: Developed to address the limitations of both the Van der Waals and Redlich-Kwong equations, it introduces a more complex dependence on temperature and volume. This equation is especially useful for predicting the properties of gases and liquids near the critical point and at high pressures. It is widely used in industries like petroleum engineering because it provides a good balance between complexity and accuracy.
Each equation varies in terms of accuracy and complexity, with the Peng-Robinson equation generally providing the most reliable results across a wide range of conditions.