Problem 90 - A Complete Resource Book in Chemistry for JEE Main

Question

For non-zero value of force of attraction between gas molecules, gas equation will be (a) \(\mathrm{PV}=\mathrm{n} \mathrm{RT}-\frac{\mathrm{n}^{2}}{\mathrm{~V}} \mathrm{a}\) (b) \(\mathrm{PV}=\mathrm{nRT}+\mathrm{nbP}\) (c) \(\mathrm{P}=\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{b}}\) (d) \(\mathrm{PV}=\mathrm{nRT}\)

Step-by-Step Solution

Answer

The answer is (a) \( PV = nRT - \frac{n^2}{V} a \).

Step 1: Identify the Ideal Gas Equation

The ideal gas law equation is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature. In the ideal gas equation, it is assumed that there is no interaction between the molecules.

Step 2: Understand the Van der Waals Equation

The Van der Waals equation corrects for the interactions between molecules (attraction and repulsion) and the volume occupied by gas molecules themselves. The Van der Waals equation is \[ \left(P + \frac{a n^2}{V^2}\right) (V - nb) = nRT \], where \( a \) represents the magnitude of the attractive forces between particles, and \( b \) accounts for the volume occupied by the gas molecules.

Step 3: Relate Parameters to the Force of Attraction

For a non-zero force of attraction, the key correction in the Van der Waals equation is the \( \frac{a n^2}{V^2} \) term, which accounts for intermolecular attractions. Rearranging the equation showcases that \( PV \) would be less than \( nRT \) if these attractions are significant, specifically \( PV = nRT - \frac{an^2}{V} \).

Step 4: Review Given Options

Review the options to identify which reflects a non-zero attractive force. The option that modifies the ideal gas law to include a correction for attraction (\( -\frac{an^2}{V} \)) is option (a): \( PV = nRT - \frac{n^2}{V}a \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law provides a simple relationship between pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the number of moles (\( n \)) of a gas. It is expressed by the formula \( PV = nRT \), where \( R \) is the universal gas constant. This law assumes that the gas molecules do not interact and the volume occupied by the gas molecules themselves is negligible. While it works well under many conditions, especially at low pressures and high temperatures, it falls short in accurately describing the behavior of gases under other conditions.

Intermolecular Forces

Intermolecular forces are the attractive and repulsive forces between molecules. These forces, albeit relatively weak compared to chemical bonds, play a crucial role in determining the physical properties of gases.

• **Attractive Forces**: Often lead to a reduction in effective pressure exerted by the gas molecules on the walls of the container.
• **Repulsive Forces**: Predominantly effective at short distances and are responsible for the physical presence of molecules.

These forces are incorporated in the Van der Waals equation, allowing for a more precise calculation of a real gas’s behavior, especially when compared to the assumptions made by the Ideal Gas Law.

Real Gases

Real gases deviate from the Ideal Gas Law primarily due to intermolecular forces and the finite size of molecules. Unlike ideal gases, real gases occupy space and experience attraction, impacting volume and pressure.
To adjust for these factors, corrections are applied via the Van der Waals equation, which introduces parameters \( a \) and \( b \).

• **\( a \):** Reflects the magnitude of intermolecular attractions. Higher values of \( a \) indicate stronger attractions.
• **\( b \):** Represents the volume occupied by gas molecules themselves.

These corrections result in an equation allowing for accurate predictions of gas behavior in non-ideal conditions, offering insights into complex interactions in various scenarios.

Gas Laws

Gas laws encompass a range of equations that describe the behavior of gases by relating measurable properties like pressure, volume, temperature, and number of moles.**Popular Gas Laws Include:**

• **Boyle's Law:** \( P \times V = \text{constant} \) (at constant \( T \) and \( n \)), illustrating an inverse relationship between pressure and volume.
• **Charles's Law:** \( V \propto T \) (at constant \( P \) and \( n \)), emphasizing the direct proportional relationship between volume and temperature.
• **Avogadro's Law:** \( V \propto n \) (at constant \( P \) and \( T \)), outlining a direct relationship between volume and number of moles.

Each gas law provides valuable insights into how gases behave under certain conditions, and together they form a comprehensive framework that helps in understanding and investigating gas behaviors in different contexts.