Problem 119 - A Complete Resource Book in Chemistry for JEE Main

Question

Mark the correct statements (a) At low pressure, the van der Waal's equation is written as $$ \left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right) \times \mathrm{V}=\mathrm{RT} $$ (b) When \(\mathrm{Z}>1\), at \(\mathrm{STP}, \mathrm{V}_{\text {real }}>\mathrm{V}_{\text {ideal }}\) (c) Mean free path of \(\mathrm{O}_{2}\) is greater than that of \(\mathrm{H}_{2}\). (d) At \(273 \mathrm{~K}\), the total kinetic energy of \(\mathrm{O}_{2}\) will be eight times that of one mole of He.

Step-by-Step Solution

Answer

Statements (a) and (b) are correct.

Step 1: Analyzing Statement (a)

The van der Waals equation is a modified version of the ideal gas law that accounts for intermolecular forces and the volume occupied by gas particles. For low pressure, the volume correction can be negligible, simplifying the equation to \[\left(P + \frac{a}{V^2}\right) \times V = RT\]. This statement is correct because at low pressures, the equation approximates to the given form.

Step 2: Analyzing Statement (b)

Compressibility factor \(Z = \frac{PV}{nRT}\) measures the deviation of a real gas from ideal behavior. When \(Z > 1\), it indicates repulsive forces are dominant, suggesting a larger observed volume, hence \(V_{\text{real}} > V_{\text{ideal}}\). Thus, this statement is correct.

Step 3: Analyzing Statement (c)

The mean free path is inversely proportional to the molecular size and directly proportional to temperature and inversely to pressure. \(\mathrm{H}_2\) is smaller in size than \(\mathrm{O}_2\), thus having a greater mean free path. This makes the statement incorrect.

Step 4: Analyzing Statement (d)

The total kinetic energy of gas particles depends only on temperature for an ideal gas. Both \(\mathrm{O}_2\) and \(\mathrm{He}\) at the same temperature will have the same kinetic energy per mole. Hence, this statement is incorrect, as the energy will not be eight times that of \(\mathrm{He}\).

Key Concepts

Ideal Gas Law

The ideal gas law is an essential principle in chemistry and physics used to describe the behavior of an ideal gas. It is expressed with the formula \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of the gas.
• \(R\) is the universal gas constant.
• \(T\) is the temperature of the gas in Kelvin.

This equation assumes that gas particles have no volume and do not experience intermolecular forces. However, this is an idealization and doesn’t account for real gas behaviors under high pressure or low temperature. The van der Waals equation, a modified version, includes these factors to provide a more accurate description. At conditions approaching low pressures and high temperatures, gases behave more ideally and the simple ideal gas law is quite sufficient.

Compressibility Factor

The compressibility factor, denoted as \(Z\), expresses how much a real gas deviates from ideal gas behavior. It is defined by the formula \[Z = \frac{PV}{nRT}\].

• When \(Z = 1\), the gas behaves ideally.
• If \(Z < 1\), the gas is more compressible than expected, often due to attractive forces between molecules.
• If \(Z > 1\), repulsive forces dominate, leading to larger volumes than predicted by the ideal gas law.

Understanding \(Z\) provides insights into molecular interactions and the conditions under which gases might deviate from ideal behavior. At standard temperature and pressure (STP), deviations can arise, and knowing \(Z\) helps chemists and engineers determine the real gas's behavior compared to the theoretical model.

Mean Free Path

Mean free path is a concept describing the average distance a molecule travels before colliding with another molecule. It relies on factors such as size, temperature, and pressure. Mathematically, it is inversely related to the size of the molecules and the pressure of the environment while being directly proportional to the temperature.

• Molecules in a gas with large mean free paths often indicate lower pressure or smaller molecular size.
• For example, \(H_2\) gas molecules, being smaller than \(O_2\) molecules, generally have a greater mean free path under similar conditions.

Comprehending mean free path is crucial for understanding gas diffusion or effusion, and the kinetics of gas molecules in various environmental conditions.

Kinetic Energy of Gases

The kinetic energy of a gas pertains to the energy possessed by gas particles due to their motion. For an ideal gas, at a given temperature, the total kinetic energy is directly proportional to the temperature in Kelvin. This is articulated in the formula: total kinetic energy per mole \(E_k = \frac{3}{2}RT\).

• All gases at the same temperature have similar average kinetic energy per particle.
• Though \(O_2\) molecules are heavier compared to \(He\), at constant temperature, their kinetic energy remains the same on a per mole basis.

This understanding is crucial when analyzing temperatures' influence on gas behavior, as it indicates that the kinetic energy of gases does not depend on their identity but solely on the temperature.