Problem 42 - A Complete Resource Book in Chemistry for JEE Main

Question

The term that accounts for intermolecular force in van der Waals equation for non ideal gas is (a) \(\mathrm{RT}\) (b) \(\mathrm{V}-\mathrm{b}\) (c) \(\left(\mathrm{P}+\alpha \mathrm{V}^{2}\right)\) (d) \((\mathrm{RT})^{-1}\)

Step-by-Step Solution

Answer

The correct answer is (c) \(\mathrm{P}+\alpha \mathrm{V}^{2}\).

Step 1: Understand van der Waals Equation

van der Waals equation for real gases is given by: \[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \] where \(P\) is the pressure, \(V_m\) is the molar volume, \(T\) is the temperature, \(R\) is the universal gas constant, \(a\) accounts for intermolecular forces, and \(b\) is a constant accounting for the volume occupied by the gas molecules.

Step 2: Identify the term for intermolecular forces

In the van der Waals equation, the term \(\frac{a}{V_m^2}\) is added to the pressure \(P\) to correct for the attractive forces between molecules. This accounts for the intermolecular forces present in a non-ideal gas.

Step 3: Match with given options

Compare the term for intermolecular forces, \(\frac{a}{V_m^2}\), with the given options: (a) \(\mathrm{RT}\), (b) \(\mathrm{V}-\mathrm{b}\), (c) \(\mathrm{P}+\alpha \mathrm{V}^{2}\), and (d) \((\mathrm{RT})^{-1}\). Option (c) is closest to \(P + \frac{a}{V_m^2}\), assuming \(\alpha\) represents \(a\) and \(\mathrm{V}^{2}\) represents \(V_m^2\).

Key Concepts

Intermolecular Forces

Intermolecular forces are the attractions between molecules that influence their behavior and properties. These forces are a key topic in the study of real gases and are distinct from the strong covalent or ionic bonds within molecules. In the context of gases, intermolecular forces become particularly significant in non-ideal conditions where real gases do not behave exactly like ideal gases.

There are several types of intermolecular forces that can affect gas behavior:

• **London Dispersion Forces:** These are weak forces that arise from temporary fluctuations in the electron distribution within molecules, leading to temporary dipoles.
• **Dipole-Dipole Interactions:** These occur between molecules that have permanent dipoles, leading to an attractive force between the positive end of one molecule and the negative end of another.
• **Hydrogen Bonds:** A special type of dipole-dipole interaction occurring when hydrogen is bonded to a highly electronegative atom like nitrogen, oxygen, or fluorine, creating an exceptionally strong force.

In the van der Waals equation, the parameter "a" is introduced to correct for these intermolecular attractions by modifying the effective pressure applied to the gas. This adjustment helps in predicting real gas behavior more accurately.

Real Gases

Real gases differ from ideal gases primarily due to the presence of intermolecular forces and the finite volume occupied by gas molecules. Understanding these differences is essential in chemical and physical studies where precise gas behavior is analyzed.

While ideal gases are described by the Ideal Gas Law, \( PV = nRT \), which assumes no volume and no intermolecular attraction, real gases require more sophisticated models like the van der Waals equation to accurately describe their behavior under various conditions.

**Characteristics of Real Gases**

• **Volume:** Real gas molecules occupy a finite volume, affecting the available space for the gas to move. This is accounted for by the "b" parameter in the van der Waals equation.
• **Intermolecular Forces:** As discussed, real gases experience attractions or repulsions between molecules, affecting their overall pressure and compressibility.
• **Behavior Under Different Conditions:** At high pressures and low temperatures, real gases exhibit more non-ideal behavior. This is where deviations from the Ideal Gas Law are most pronounced.

The behavior of real gases is critical for applications in fields like engineering, meteorology, and materials science, where accurate predictions of gas behavior influence design and operation.

Non-Ideal Gas Behavior

Non-ideal gas behavior is observed when gases deviate from the assumptions of the Ideal Gas Law, particularly at high pressures and low temperatures. These deviations arise due to intermolecular forces and molecular volume, which become more significant under such conditions.

**Key Factors of Non-Ideal Behavior**

• **High Pressure:** At high pressures, gas molecules are forced closer together, enhancing intermolecular attractions or repulsions. This effect results in pressure being different from what is predicted by the Ideal Gas Law.
• **Low Temperature:** Decreased kinetic energy at low temperatures means that intermolecular forces have a more substantial effect on gas behavior, increasing deviations from ideal predictions.

In the van der Waals equation, modifications are made to account for these non-ideal behaviors:

• The term \( P + \frac{a}{V_m^2} \) corrects for intermolecular attractions, effectively reducing the measured pressure to reflect the actual force experienced by the gas molecules.
• The volume correction \( V_m - b \) is adjusted to account for the finite volume of molecules, ensuring that the space they occupy is not counted as available volume for motion.

Understanding non-ideal gas behavior is significant for industries and sciences where exact knowledge of gas properties can influence processes such as gas compression, liquefaction, and chemical reactions.