Problem 173 - Master Resource Book in JEE Main Physics

Question

The value of \(\frac{p V}{T}\) for one mole of an ideal gas is nearly equal to \(\quad\) [BVP Engg. 2007] (a) \(2 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (b) \(8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (c) \(4.2 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (d) \(2 \mathrm{cal} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\)

Step-by-Step Solution

Answer

The value of \( \frac{p V}{T} \) is (b) 8.3 J mol\(^{-1}\) K\(^{-1}\).

Step 1: Understand the Ideal Gas Law

The ideal gas law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature. For one mole of gas, \( n = 1 \), so the equation simplifies to \( PV = RT \).

Step 2: Isolate \( \frac{pV}{T} \)

Divide both sides of the equation \( PV = RT \) by \( T \) to isolate \( \frac{pV}{T} \), which gives us \( \frac{pV}{T} = R \).

Step 3: Recall the value of \( R \)

The universal gas constant \( R \) is approximately \( 8.314 \) J mol\(^{-1}\) K\(^{-1}\).

Step 4: Match with given options

Compare the calculated value \( 8.3 \) J mol\(^{-1}\) K\(^{-1}\) to the provided options. The closest match is option (b) \( 8.3 \) J mol\(^{-1}\) K\(^{-1}\).

Key Concepts

Universal Gas Constant

The Universal Gas Constant, often denoted by the symbol \( R \), is a fundamental component in the Ideal Gas Law equation. It acts as the bridge linking the behavior of gases with the principles of thermodynamics. The constant \( R \) is vital for calculating various properties of an ideal gas, such as volume, pressure, and temperature under set conditions. Its value is approximately \( 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \). This means that for every mole of an ideal gas under specified conditions, \( 8.314 \) joules of energy are required to change the temperature by one Kelvin.

• Essential for computations involving gas laws.
• Connects energy scales to temperature and pressure measures in gas behavior.
• Key in many chemical equations and involved in calculations related to energy required for reactions.

Understanding \( R \) unlocks various pathways in studies related to thermodynamics and chemical reactions, influencing how energy changes and interactions are quantified.

Mole Concept

The Mole Concept is foundational in chemistry for quantifying the amount of substance. One mole corresponds to \( 6.022 \times 10^{23} \) entities (like atoms or molecules), known as Avogadro's number. In the context of the Ideal Gas Law, 'moles' (\( n \)) specify how many gas entities are involved.

• Allows chemists to count particles in a tractable way using macroscopic measurements.
• Facilitates conversions between atomic scale measures and laboratory scale results.
• Moles link the mass of a substance to the number of particles it contains, bridging micro and macro scales.

When performing calculations involving gases, knowing the number of moles helps in predicting and understanding gas behavior under different conditions, showcasing its centrality in chemistry and physics.

Thermodynamics

Thermodynamics is the science of energy transformation and transfer. It explains how energy exchanges occur in chemical processes, including those involving gases. In the realm of the Ideal Gas Law, thermodynamics aids in comprehending the relationship between different state functions such as volume, pressure, and temperature.

• Key principles include energy conservation and entropy, dictating the energy flow and system changes.
• The laws of thermodynamics describe how energy is converted during physical and chemical processes.
• In gas laws, thermodynamics elucidates how work and heat interact with the gas system.

By understanding thermodynamic principles, one can predict how a gas will respond to changes in its environment, granting insight into the intrinsic chemical and physical nature of gases.

Pressure-Volume Relationship

The Pressure-Volume Relationship is a critical aspect of gas behavior. According to Boyle's Law, at a constant temperature, the pressure of a gas is inversely proportional to its volume (\( P \propto \frac{1}{V} \)). In the framework of the Ideal Gas Law, this relationship helps to determine how gases expand or compress when subjected to changes in pressure or volume.

• Illustrates the squeeze-and-stretch behavior of gases when pressure or volume changes.
• Vital for understanding compressibility and expansion characteristics of gases.
• Grounded in basic principles that help predict the effect of mechanical stresses on a gas.

Grasping the pressure-volume relationship lays the groundwork for further exploration in topics such as fluid dynamics and material science, contributing to innovations in engineering and technology.