Question

One mole of each gases \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) with van der Waal's constant \(\left(\mathrm{atm} 1^{2} \mathrm{~mol}^{-2}\right)\) \(1.348,6.823,4.390\) and \(2.438\), respec- tively, are kept separately in four different vessels of equal volumes at identical temperature. Their pressures are observed to be \(P_{A}, P_{\mathrm{B}}, P_{\mathrm{C}}\) and \(P_{\mathrm{D}}\), respectively. On the basis of this data alone, the order of pressure of gases is (assume other van der Waal's constant to be nearly same for all gases) (a) \(P_{\mathrm{A}}

Short answer

The correct order of pressures from lowest to highest, based on the given van der Waals constants, is: \(P_{\text{B}} < P_{\text{C}} < P_{\text{D}} < P_{\text{A}}\).

Step-by-step solution

Understand van der Waals Equation

The van der Waals equation for real gases is given by \((P + \frac{an^2}{V^2})(V - nb) = nRT\), where \(P\) is the pressure of the gas, \(n\) is the number of moles, \(V\) is the volume, \(R\) is the ideal gas constant, \(T\) is the temperature, and \(a\) and \(b\) are van der Waals constants. Since the volumes, temperature and \(b\) constant are the same for all gases and each gas has one mole, the equation simplifies to \(P = \frac{nRT}{V} - \frac{a}{V^2}\).

Focus on the Relevant part of the Equation

Because conditions such as volume, temperature, and the moles of the gas are constant, and we're assuming \(b\) to be nearly same for all gases, we focus only on the \(\frac{a}{V^2}\) term. This term is subtracted from the pressure equation, indicating that the larger the value of \(a\), the more the pressure of the real gas will deviate from the ideal behavior.

Analyze Inverse Relationship

As the value of \(a\) increases, the corrective pressure term \(\frac{a}{V^2}\) also increases, leading to a greater reduction in pressure from the ideal case. Thus, the higher the value of \(a\), the lower the observed pressure, assuming other conditions are constant.

Compare the van der Waals constants

By comparing the given van der Waals constants, \(a_A = 1.348\), \(a_B = 6.823\), \(a_C = 4.390\), and \(a_D = 2.438\), we can see that \(a_B\) is the highest, followed by \(a_C\), then \(a_D\), and finally \(a_A\) as the smallest.

Determine Pressure Order

Following the inverse relationship, the order of pressures from lowest to highest will be the reverse of the order of their van der Waals 'a' constants. Hence, the order of pressure will be: \(P_B < P_C < P_D < P_A\).

Key concepts

Real Gases Behavior

Unlike ideal gases, which follow the ideal gas law precisely, real gases display complexities that require modification of the law to account for their actual behavior. The ideal gas law, \(PV=nRT\), presumes no forces between gas particles and assumes that the volume occupied by the gas particles is negligible compared to the container volume.

However, real gases deviate from this simplicity due to particle interactions and finite particle size, especially under high pressure or low temperature conditions where particles are closer together. These deviations include attraction between gas molecules and the volume occupied by the molecules themselves. The van der Waals equation is an amendment that accounts for these factors, bringing real-world observations into closer alignment with theoretical predictions.

To grasp the concept thoroughly, imagine that intermolecular forces cause the gases to exert less pressure on the container walls than predicted by the ideal gas law. Also, because gas molecules occupy space, there's less room for movement, further affecting the pressure. These considerations are vital when working with real gases in a laboratory or industrial setting.

Van der Waals Constants

The van der Waals equation introduces two constants, \(a\) and \(b\), which provide corrections for the pressure and volume of a gas, respectively. The constant \(a\) accounts for intermolecular attractions; larger values of \(a\) indicate stronger attractive forces between particles. The constant \(b\) corrects for the volume occupied by gas molecules, with larger values of \(b\) suggesting larger particles.

In the context of textbook exercises, comparing the \(a\) values for different gases can illuminate their predicted behaviors. For the sake of simplicity, if we consider four different gases with varying \(a\) values, a higher \(a\) value would generally lead to a more significant reduction in the pressure exerted by a gas, illustrating a greater deviation from ideal gas behavior. Students should remember that these constants are unique to each gas and are experimentally determined, reflecting the unique properties of different molecules.

Understanding these constants helps in predicting how similar a real gas is to an ideal gas under certain conditions, crucial when precise measurements are necessary in applications like chemical engineering or pharmaceuticals.

Inverse Pressure Relationship

The inverse pressure relationship in the van der Waals equation captures the physical reality that as the attractive forces between gas molecules increase, represented by an increase in the \(a\) constant, the observed pressure decreases from what would be expected if the gas were ideal. This is because these intermolecular forces reduce the kinetic energy of the particles hitting the container walls, thus reducing the pressure.

In an exercise where different gases have the same temperature, volume, and moles, and their \(b\) constants are approximately equal, the pressure of each gas is primarily influenced by the \(a\) constant. This is where the inverse pressure relationship becomes evident. With all other factors being constant, a higher \(a\) constant suggests a larger deduction from ideal pressure, resulting in a lower actual pressure. This relationship is pivotal in determining the behavior of a gas and is a fundamental concept when learning about real gases.

Through exercises involving the van der Waals equation, students can practice using this relationship to predict the pressures of real gases, comparing them to one another and enhancing their understanding of the complex interplay between microscopic forces and macroscopic measurements.