Question

In the corrections made to ideal gas equation for real gases, the reductions in pressure due to forces of attractions between the molecules is directly peoportional to (a) \(\frac{n}{V}\) (b) \(\frac{n^{2}}{V^{2}}\) (c) \(\boldsymbol{V}-\boldsymbol{n} b\) (d) \(n b\)

Short answer

\frac{n^{2}}{V^{2}}

Step-by-step solution

Understanding the Van der Waals Equation

The corrections made to the ideal gas equation for real gases are accounted for by the Van der Waals equation, which adjusts pressure and volume to account for molecular attractions and the finite size of molecules. The Van der Waals equation is given by: \[\left(P + \frac{a n^{2}}{V^{2}}\right)\left(V-nb\right)= nRT\] Where: - P is the corrected pressure - V is the corrected volume - n is the number of moles - T is the temperature - R is the ideal gas constant - a and b are Van der Waals constants

Identifying Pressure Correction Term

From the Van der Waals equation, the term that accounts for the reduction in pressure due to forces of attraction is the term added to the pressure, which is \[\frac{a n^{2}}{V^{2}}\] This term is directly proportional to the square of the number of moles divided by the square of the volume, representing the interactions between molecules.

Key concepts

Real Gases

When we study gases in our chemistry classes, we typically begin with an idealized model known as an 'ideal gas.' This model assumes that the gas particles do not interact with each other and take up no space. However, in reality, these assumptions are not true for real gases. The behavior of real gases deviates from the ideal model, especially at high pressures and low temperatures, where the volume of the gas particles and the forces between them can no longer be ignored.

Deviations from Ideal Behavior

• Gas particles have their own volume, which becomes significant at high pressures, reducing the space available for particles to move.
• Intermolecular forces begin to play a role, particularly at low temperatures where the particles are closer together.

Understanding real gases requires corrections to the ideal gas law, which can be achieved through the Van der Waals equation, a more accurate representation for these conditions. This equation makes it possible to predict the behavior of gases under a variety of conditions, accounting for the volume occupied by the gas particles and the forces of attraction and repulsion between them.

Intermolecular Forces

Intermolecular forces are the forces of attraction and repulsion that exist between molecules. They play a crucial role in determining the physical properties of substances, including gases. In the context of gases, these forces affect how molecules interact with one another when they are in close proximity.

Types of Intermolecular Forces

• Dispersion forces: These occur between all molecules, induced by the movement of electrons that create temporary dipoles.
• Dipole-dipole interactions: These occur between polar molecules, where the positive end of one molecule attracts the negative end of another.
• Hydrogen bonds: A special type of dipole-dipole interaction that occurs when hydrogen is bonded to a highly electronegative atom.
• Ion-dipole forces: Occur between ions and polar molecules and are important in solutions.

The strength of these forces can affect the volume and pressure of a gas. When gas particles are pushed closer together, the attractive forces between them can cause the pressure to be less than what we would expect based on the ideal gas law. This is why corrections for real gases are essential for a more accurate description of their behavior.

Correction to Ideal Gas Law

The ideal gas law, represented by the equation PV = nRT, is a foundation of gas laws and assumes that gas particles are point particles in constant, random motion with no interactions between them. In real-life scenarios, this assumption often does not hold, particularly under conditions of high pressure and low temperature, where the volume of gas particles and their intermolecular forces can significantly affect the gas behavior.

The Van der Waals EquationTo account for these non-ideal behaviors, scientists such as Johannes Diderik van der Waals introduced corrections to the ideal gas law, leading to the Van der Waals equation: \begin{align*} \left(P + \frac{a n^{2}}{V^{2}}\right)\left(V-nb\right)&= nRT\end{align*}

• a: Represents the correction for intermolecular forces. A higher 'a' value indicates stronger intermolecular attraction.
• b: Accounts for the finite volume of gas particles. A larger 'b' value reflects a larger particle size.

The term \(\frac{a n^{2}}{V^{2}}\) is critical as it corrects the pressure to reflect the effect of intermolecular attractions between the gas particles, causing the observed pressure to be lower. The reduction in pressure is indeed directly proportional to the square of the number of moles per unit volume—represented by \(\frac{n^{2}}{V^{2}}\)—demonstrating the direct impact that intermolecular forces have on gas behavior.