Calculate The Compressibility Factor (Z) Of CH₄ If The Density Is 0.25 Gm/ml At 200 K And 500 Atm. (R = 0.08 L-atm/K-mol)

#h1 Understanding Compressibility Factor of CH₄ at 200 K and 500 atm

In the realm of thermodynamics, the compressibility factor (Z) serves as a crucial indicator of a real gas's deviation from ideal gas behavior. This comprehensive exploration delves into calculating the compressibility factor of methane (CH₄) under specific conditions: a temperature of 200 K and a pressure of 500 atm. We will leverage the given density of 0.25 gm/ml and the ideal gas constant (R = 0.08 L-atm/K-mol) to navigate this calculation. This article will provide a step-by-step guide, ensuring clarity and a thorough understanding of the underlying principles. Accurately determining the compressibility factor (Z) is essential for various engineering applications, particularly in the design and operation of processes involving high-pressure gases. Understanding how real gases deviate from ideal behavior allows for more precise calculations in chemical engineering, materials science, and other fields. This introduction sets the stage for a detailed analysis, emphasizing the practical significance of the compressibility factor in real-world scenarios. By meticulously examining each step, we aim to equip readers with the knowledge to confidently tackle similar problems and appreciate the nuances of gas behavior under extreme conditions. The importance of understanding gas behavior under non-ideal conditions cannot be overstated, especially in industrial settings where precision and safety are paramount. This article is tailored not only for students and academics but also for professionals seeking a robust understanding of these principles. We will also discuss the implications of a compressibility factor (Z) deviating from unity, shedding light on the intermolecular forces at play within the gas. By carefully considering all these aspects, we can gain a deeper appreciation for the complexities of gas behavior and its impact on various applications.

Decoding the Compressibility Factor (Z)

The compressibility factor (Z) is a dimensionless quantity that quantifies the deviation of a real gas from ideal gas behavior. For an ideal gas, Z = 1. However, real gases exhibit deviations from ideality, especially at high pressures and low temperatures. Values of Z greater than 1 indicate that the gas is less compressible than an ideal gas, implying that repulsive forces dominate. Conversely, values of Z less than 1 suggest that the gas is more compressible than an ideal gas, indicating that attractive forces prevail. Understanding the compressibility factor (Z) is paramount in chemical engineering and thermodynamics, as it directly impacts the accuracy of calculations involving gas properties and behavior. The ideal gas law, PV = nRT, provides a foundational understanding of gas behavior, but it often falls short when applied to real-world conditions where intermolecular interactions become significant. The compressibility factor effectively corrects the ideal gas law, allowing for more precise predictions of gas volume, pressure, and temperature relationships. This correction is particularly critical in industrial processes where gases are often subjected to extreme conditions. Furthermore, the compressibility factor (Z) provides insights into the molecular interactions within a gas. Deviations from unity reveal the extent to which intermolecular forces influence the gas's behavior. A deep understanding of these forces is crucial for designing efficient and safe processes involving gas storage, transport, and reactions. In this context, we will explore how to calculate the compressibility factor (Z) for methane (CH₄) under specific conditions, providing a practical application of these theoretical concepts. This will involve using the given density, temperature, and pressure to determine the molar volume, which is a key parameter in the compressibility factor calculation. By meticulously following the steps outlined in this article, readers will gain a solid grasp of both the theoretical underpinnings and the practical application of the compressibility factor (Z).

Step-by-Step Calculation of Z for CH₄

To determine the compressibility factor (Z) for methane (CH₄) at 200 K and 500 atm, we will follow a step-by-step approach. This ensures a clear understanding of each stage in the calculation. First, we need to calculate the molar volume (V_m) of CH₄ using the given density (ρ = 0.25 gm/ml). The molar mass of CH₄ is approximately 16 g/mol. We can convert the density from gm/ml to g/L by multiplying by 1000, giving us a density of 250 g/L. The molar volume is then calculated by dividing the molar mass by the density: V_m = Molar Mass / Density = 16 g/mol / 250 g/L = 0.064 L/mol. Next, we apply the compressibility factor equation, which modifies the ideal gas law: Z = (P * V_m) / (R * T). Here, P is the pressure (500 atm), V_m is the molar volume (0.064 L/mol), R is the ideal gas constant (0.08 L-atm/K-mol), and T is the temperature (200 K). Substituting these values into the equation, we get: Z = (500 atm * 0.064 L/mol) / (0.08 L-atm/K-mol * 200 K). Performing the calculation, Z = 32 / 16 = 2. This result indicates that the compressibility factor (Z) for methane under these conditions is 2. This significant deviation from 1 highlights that methane behaves non-ideally at 200 K and 500 atm. The high pressure and relatively low temperature cause significant intermolecular interactions, making the ideal gas law an inadequate approximation. The calculated compressibility factor (Z) of 2 implies that methane is less compressible than an ideal gas under these conditions, suggesting that repulsive forces between methane molecules are dominant. This step-by-step calculation underscores the importance of using the compressibility factor (Z) to accurately predict gas behavior under non-ideal conditions, especially in industrial applications where precise calculations are crucial for safety and efficiency. By breaking down the calculation into manageable steps, we ensure clarity and facilitate a deeper understanding of the underlying principles.

Implications of Z = 2

A compressibility factor (Z) of 2 for methane (CH₄) at 200 K and 500 atm carries significant implications regarding its behavior and intermolecular interactions. This value, substantially greater than 1, indicates that methane deviates significantly from ideal gas behavior under these conditions. Specifically, it suggests that methane is less compressible than an ideal gas, meaning that the actual volume occupied by the gas is greater than what would be predicted by the ideal gas law. This deviation arises from the repulsive forces between methane molecules, which become more pronounced at high pressures and relatively low temperatures. At 500 atm, the methane molecules are forced into close proximity, enhancing the repulsive forces and causing the gas to resist compression. The compressibility factor (Z) provides a quantitative measure of these deviations, allowing for more accurate predictions of gas behavior in industrial processes. In practical terms, a Z value of 2 means that calculations based on the ideal gas law would underestimate the volume occupied by methane by a factor of two. This has crucial implications for the design and operation of high-pressure equipment, such as pipelines, storage tanks, and reactors. Engineers must account for the non-ideal behavior of gases like methane to ensure safety and efficiency. Furthermore, the compressibility factor (Z) offers insights into the intermolecular forces at play. The dominant repulsive forces suggest that the molecules are effectively