Question

Show that the constant \(a\) in the van der Waals equation can be written in units of \(\frac{\text { energy per particle }}{\text { particle density }}\)

Short answer

The constant \(a\) in the van der Waals equation can be written in units of \( \frac{\text { energy per particle }}{\text { particle density }} \), as it demonstrates the inter-particle attraction in a gas. This is shown through understanding the units involved in the equation and rearranging them to isolate and represent \(a\) in the desired units.

Step-by-step solution

Identify Parameters

We have the variables P (Pressure), V (Volume), n (number of particles), T (Temperature), and R (Gas Constant). Note that the two constants in the equation are \(a\) which accounts for the particle-particle attraction, and \(b\) making allowance for the finite size of the particles.

Understanding Units

The unit of \(a\) can be obtained from the equation \(P+a(n/V)^2\). The unit of \(P\) is \(\text { energy per unit volume }\) (E/V), and the unit of \(n/V\) is \(\text { particle density }\). Thus the unit of \((n/V)^2\) is \(\text { particle density }^2\), and therefore the unit of \(a\) must satisfy \(\text { energy per unit volume }=a * \text { particle density }^2\), which can be manipulated to show \(\text { energy per particle }= a * \(\text { particle density }\).

Solve for Unit of a

Rearranging the equation \(\text { energy per particle } = a * \(\text { particle density }\) for \(a\) gives us the required unit for \(a\), which is, \(a = \frac{\text{energy per particle}}{\text{particle density}}

Key concepts

Particle-Particle Attraction

In the context of the van der Waals equation, particle-particle attraction represents the forces that particles exert on each other in a gas. Unlike an ideal gas where particles are assumed not to attract or repel each other, in real gases, these interactions are significant and can affect the physical properties of the gas. The van der Waals equation incorporates a term represented by the constant 'a' to account for these attractive forces.

Understanding how this attraction alters the behavior of gases is crucial for accurately predicting how a real gas will react under different conditions of pressure and temperature. The presence of attraction between particles tends to decrease the pressure of a gas because particles are drawn together, reducing the collision frequency with the container walls. This is a key concept in explaining why real gases deviate from the ideal gas law at high pressures or low temperatures, where these attractions become more pronounced.

The term 'a' in the van der Waals equation mathematically corrects for these attractions by adjusting the pressure in the ideal gas equation. This adjustment is necessary to match the observed behavior of real gases and is proportional to the square of the particle density, reflecting how attractive forces increase as particles become more concentrated.

Particle Density

The concept of particle density is fundamental in describing the physical state of a gas. It is defined as the number of particles per unit volume, often represented as n/V where 'n' stands for the number of particles, and 'V' denotes the volume the gas occupies. In the van der Waals equation, particle density comes into play when we discuss both the attraction between particles and the volume that particles themselves occupy.

A higher particle density implies more particles in a given space, which intensifies the interaction between them, including both attraction and repulsion. The van der Waals constant 'b' accounts for the volume occupied by the gas particles themselves, indicating that actual volume available for particle movement is less than the measured volume of the container.

Meanwhile, the constant 'a' interacts with the squared particle density to correct the pressure for attraction forces. Thus, particle density is a critical factor in the van der Waals equation, influencing both pressure and volume, which are pivotal in understanding the behavior of real gases.

Pressure-Volume-Temperature Relationship

The pressure-volume-temperature relationship is a cornerstone of thermodynamics, depicting how these three state variables influence one another in gases. The van der Waals equation is a modified version of the ideal gas law that accounts for real gas behavior by incorporating particle-particle attraction and the finite size of particles.

This relationship in the ideal gas law is expressed as PV=nRT, where P is pressure, V is volume, T is temperature, n is the number of moles, and R is the universal gas constant. However, this equation assumes that gases have no intermolecular forces and the particles take up no space — assumptions that fail under high pressure and low temperature.

The van der Waals equation refines this relationship by including two additional parameters, 'a' and 'b', which represent the effect of intermolecular attractions and the volume occupied by gas molecules, respectively. The equation is written as \(P + \frac{an^2}{V^2}\)(V - nb) = nRT. By using this equation, we're better able to predict real gas behaviors by accounting for the effects of particle density and interaction forces on the pressure-volume-temperature relationship, which is essential for applications involving gases in various fields of science and industry.