Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(Z\) is a compressibility factor, van der Waals equation at low pressure can be written as: [2014] (a) \(\mathrm{Z}=1-\frac{\mathrm{Pb}}{\mathrm{RT}}\) (b) \(\mathrm{Z}=1+\frac{\mathrm{Pb}}{\mathrm{RT}}\) (c) \(\mathrm{Z}=1+\frac{\mathrm{RT}}{\mathrm{Pb}}\) (d) \(Z=1+\frac{a}{V R T}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(Z = 1 + \frac{Pb}{RT}\).

Step 1: Understand Van der Waals Equation

The van der Waals equation is a modification of the ideal gas law, used to account for the finite size of molecules and the intermolecular forces between them. It is written as: \[ \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \] where \(P\) is pressure, \(V_m\) is the molar volume, \(T\) is the temperature, \(R\) is the universal gas constant, \(a\) accounts for the attraction between particles, and \(b\) accounts for the volume occupied by the gas particles.

Step 2: Simplify the Equation for Low Pressure

At low pressure, the effect of intermolecular forces is less significant, which means that the term \(\frac{a}{V_m^2}\) becomes negligible. Thus, at low pressure, the equation simplifies to: \[ P(V_m - b) = RT. \]

Step 3: Derive the Compressibility Factor Expression

The compressibility factor \(Z\) is defined as \(Z = \frac{PV_m}{RT}\). From the simplified equation \(P(V_m - b) = RT\), we can write \(PV_m = P(V_m - b) + Pb = RT + Pb\). Thus, \(Z = \frac{RT + Pb}{RT} = 1 + \frac{Pb}{RT}.\)

Step 4: Identify the Correct Option

From the derived expression \(Z = 1 + \frac{Pb}{RT}\), we match this with the given options: (a) \(Z = 1 - \frac{Pb}{RT}\), (b) \(Z = 1 + \frac{Pb}{RT}\), (c) \(Z = 1 + \frac{RT}{Pb}\), and (d) \(Z = 1 + \frac{a}{VRT}\). The correct option is (b), \(Z = 1 + \frac{Pb}{RT}\).

Key Concepts

Compressibility Factor

The compressibility factor, often denoted as \(Z\), is a crucial concept in understanding the behavior of gases. It is defined by the equation \(Z = \frac{PV}{RT}\), where \(P\) is pressure, \(V\) is volume, \(R\) is the ideal gas constant, and \(T\) is temperature. The compressibility factor indicates how much a real gas deviates from the behavior predicted by the Ideal Gas Law.
If \(Z = 1\), the gas behaves like an ideal gas. This means that the gas particles are not experiencing significant intermolecular forces or occupying a significant volume, which is assumed in ideal conditions.
It is common to find \(Z < 1\) when gases are under high pressure or low temperature, indicating that attractive forces between molecules dominate. Conversely, \(Z > 1\) often occurs when repulsive forces are significant, pushing molecules apart due to high energy or low volume.

• Understanding \(Z\): Helps quantify how real gases differ from ideal behavior.
• Practical uses: Engineers and scientists use \(Z\) to correct the ideal gas calculations for better accuracy in real-world applications.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics, connecting pressure, volume, and temperature of gases in a simple formula: \(PV = nRT\). Here, \(P\) represents the pressure, \(V\) is the volume, \(n\) is the number of moles of the gas, \(R\) is the gas constant, and \(T\) is the temperature measured in Kelvin.
The law provides a simplified description of gas behavior under certain ideal conditions, primarily when intermolecular forces and the volume of gas molecules are negligible.
While the Ideal Gas Law is an approximation, it proves to be sufficiently accurate under conditions of low pressure and high temperature. These conditions indicate an environment where intermolecular forces and molecular volumes have minimal impact.
However, deviations arise when gases are at high pressure or low temperature. The molecules come closer, thus increasing the significance of the volume and forces, which leads to the necessity of modifications such as the Van der Waals equation.

• Simplicity: Offers a straightforward calculation with minimal parameters.
• Limitations: Accurate only under certain conditions; real gases require adjustments.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between neighboring molecules. These forces are vital for understanding gas deviations from ideal behavior, regularly impacting the prediction of gas properties using the Ideal Gas Law.
Typically, there are three primary kinds of intermolecular forces:

• Dispersion Forces: Also known as London forces, these are the weakest, occurring by temporary changes in electron density.
• Dipole-Dipole Forces: Present in polar molecules with permanent dipoles aligning positive ends to negative ends.
• Hydrogen Bonding: A strong type of dipole-dipole interaction, occurring specifically when hydrogen is bonded to electronegative atoms like oxygen, nitrogen, or fluorine.

These forces become more significant at low temperatures and high pressures. In such cases, molecules come closer, making the volume of gas and these attractions non-negligible.
The Van der Waals equation accounts for these forces by incorporating constants \(a\) and \(b\), which correct the Ideal Gas Law. The constant \(a\) adjusts for the force of attraction between molecules, while \(b\) reflects the volume occupied by gas particles.
Understanding these forces allows for more accurate predictions of how real gases will behave, particularly important in chemical and industrial processes.