Question

The plot of \(Z\) versus \(P\) for a gas at \(0^{\circ} \mathrm{C}\) is shown. Explain the causes of the negative deviation from ideal behavior at lower pressures and the positive deviation from ideal behavior at higher pressures.

Short answer

Negative deviations occur due to intermolecular attractions, while positive deviations arise from finite molecular volume.

Step-by-step solution

Understanding Ideal Gas Behavior

An ideal gas is modeled by the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is moles of gas, \(R\) is the ideal gas constant, and \(T\) is temperature. Under ideal conditions, gas particles are assumed to occupy no volume and exert no intermolecular forces on each other.

Negative Deviation at Low Pressure

At low pressures, gas molecules have larger distances between them, which results in weak attractive forces becoming significant. This leads to lower pressures than expected, as some energy is used to overcome these forces, leading to a negative deviation from ideality.

Positive Deviation at High Pressure

At high pressures, gas molecules are forced close together, and their finite volume becomes significant. As molecules occupy more space relative to the container size, they prevent other molecules from occupying that space, leading to a higher pressure than predicted by ideal behavior, hence a positive deviation.

Van der Waals Equation

The deviations from ideal behavior are explained by the Van der Waals equation, which corrects the ideal gas law by accounting for the volume occupied by gas molecules (\(b\) term) and the intermolecular forces (\(a\) term). This equation is given by \[(P + \frac{a}{V^2})(V - b) = nRT.\]

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental concept in chemistry and physics, represented by the equation \(PV = nRT\). This equation describes the relationship between pressure \(P\), volume \(V\), the number of moles \(n\), the ideal gas constant \(R\), and temperature \(T\). In an ideal scenario, which this equation represents, gas molecules are assumed to be point particles that do not occupy any space and do not interact with each other through intermolecular forces.

• Gas particles move in constant motion.
• Collisions between gas molecules are perfectly elastic.
• These assumptions lead to predictable behavior that can be easily modeled.

However, real gases deviate from this ideal behavior under certain conditions, such as low or high pressures. That's where concepts like negative and positive deviation come into play.

Negative Deviation

Negative deviation from ideal gas behavior occurs primarily at low pressures. In this context, the distances between gas molecules increase, and weak attractive forces, often termed as "intermolecular forces," become significant. These forces pull the gas molecules closer together, reducing the overall pressure exerted by the gas.
This deviation means that at lower pressures, gases behave less ideally because their actual pressure is lower than what the ideal gas law would predict. This is a result of the attractive forces using up part of the kinetic energy that would otherwise contribute to movement and collisions of gas particles.

• Lower than expected pressure due to intermolecular attraction.
• Significant at low pressures where molecules are more spaced out.
• Commonly observed with gases like CO₂ and NH₃.

Positive Deviation

Positive deviation occurs at high pressures where real gases deviate significantly from ideal behavior. At these pressures, gas molecules are packed closely together, and the volume they occupy becomes relevant. Unlike in the ideal gas model, real gas molecules have a finite size, and their volumes cannot be ignored.
The crowded conditions prevent gas molecules from moving freely, leading to a larger observed pressure than what the ideal gas law predicts. This is because their finite volume restricts the space available for movement, causing an increase in collisions and, consequently, pressure.

• Gas particles' own volume reduces the available space within the container.
• Higher pressure than predicted due to limited space.
• Very evident in gases at conditions close to liquefaction.

Intermolecular Forces

Intermolecular forces are the attractive or repulsive forces between molecules. They are crucial in determining the physical properties and behavior of substances, particularly gases. These forces become especially noticeable in non-ideal gas conditions, altering their expected behavior as predicted by the ideal gas law.
Types of intermolecular forces include:

• Van der Waals forces: weak interactions that include London dispersion forces and dipole-dipole interactions.
• Hydrogen bonds: a stronger type of dipole-dipole attraction involving hydrogen atoms bonded to electronegative elements like oxygen or nitrogen.

In gases, these forces are generally minimal due to the large distances between particles. However, under conditions like low temperatures or high pressures, intermolecular forces can influence the gas’s pressure, volume, and temperature quite significantly, accounting for deviations from ideal behavior.