Problem 12 - A Complete Resource Book in Chemistry for JEE Main

Question

Internal energy (E) and pressure of a gas of unit volume are related as (a) \(\mathrm{P}=\frac{2}{3} \mathrm{E}\) (b) \(\mathrm{P}=\frac{3}{2} \mathrm{E}\) (c) \(\mathrm{P}=\frac{\mathrm{E}}{2}\) (d) \(\mathrm{P}=2 \mathrm{E}\)

Step-by-Step Solution

Answer

The correct option is (a): \(P = \frac{2}{3} E\).

Step 1: Understanding the Relationship in Ideal Gas

The internal energy, E, of a monoatomic ideal gas is given by the formula \( E = \frac{3}{2} nRT \), where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. The pressure, \( P \), is expressed as \( P = \frac{nRT}{V} \). Substituting \( V = 1 \) for unit volume simplifies the expression of pressure to \( P = nRT \).

Step 2: Expressing Pressure in Terms of Internal Energy

From the equation of internal energy \( E = \frac{3}{2} nRT \), solve for \( nRT \): \( nRT = \frac{2E}{3} \). This represents the pressure \( P \) for one mole of gas of unit volume.

Step 3: Identifying the Correct Equation

We identified that \( P = nRT = \frac{2E}{3} \). Thus, the relationship between pressure \( P \) and internal energy \( E \) for a unit volume of gas is given by \( P = \frac{2}{3}E \).

Step 4: Selecting the Correct Option

By comparing the derived equation \( P = \frac{2}{3}E \) with the provided options, it matches option (a). Therefore, option (a) is the correct answer.

Key Concepts

Internal Energy of Gases

The internal energy of a gas is a crucial concept in thermodynamics. It refers to the total energy stored within the gas molecules due to their movement and interactions.
In the case of an ideal gas, the internal energy is purely a function of temperature. For a monoatomic ideal gas, this energy is represented using the formula \( E = \frac{3}{2} nRT \).

• Here, \( n \) is the number of moles.
• \( R \) is the universal gas constant with a value of 8.314 J/(mol·K).
• \( T \) is the absolute temperature measured in Kelvin.

The internal energy is directly proportional to temperature. This means that as temperature increases, so does the internal energy. The factor \( \frac{3}{2} \) is specific to monoatomic gases, which only possess translational components of kinetic energy.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of an ideal gas. It combines three important gas properties: pressure, volume, and temperature.
Mathematically, it is expressed as \( PV = nRT \). This equation helps us understand how changes in one property affect others for a fixed amount of gas.

• \( P \) stands for pressure, typically measured in Pascals (Pa).
• \( V \) is the volume of the gas, usually in cubic meters (m³).
• \( n \) represents the number of moles of gas.
• \( R \) is again the universal gas constant.
• \( T \) is the temperature in Kelvin.

The Ideal Gas Law is extremely helpful when dealing with gases at relatively low pressures and high temperatures, where they behave most ideally. It allows scientists and engineers to predict gas behavior under various conditions.

Pressure-Volume Relationship

In the context of gases, understanding the pressure-volume relationship is essential. This relationship stems from the Ideal Gas Law and is exemplified in Boyle's Law. Boyle's Law states that for a fixed amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume.
Mathematically, this is expressed as \( P \propto \frac{1}{V} \) when \( T \) and \( n \) are constant.
This means:

• If the volume increases, the pressure decreases.
• If the volume decreases, the pressure increases.

In the context of thermodynamics, when dealing with the unit volume (\( V = 1 \)), the behavior becomes simpler as seen in the example where we examine the pressure \( P = nRT\) in a unit volume scenario. This simplifies understanding of how internal energy translates into pressure in monoatomic gases.

Monoatomic Ideal Gas

A monoatomic ideal gas consists of particles that are single atoms, such as Helium or Neon gas. These have unique properties due to their simple structure.
Unlike diatomic or polyatomic gases, monoatomic gases only have translational modes of motion. This simplicity leads to different thermodynamic behaviors.

• The internal energy for monoatomic gases is solely kinetic due to translational movement.
• As a result, the internal energy formula is \( E = \frac{3}{2} nRT \), focusing on their translational kinetic energy.
• Monoatomic gases perfectly illustrate many ideal gas properties due to lack of molecular vibrations or rotations.

These characteristics make monoatomic ideal gases excellent models for studying fundamental gas laws, such as the relationship between pressure and internal energy as seen in ideal gas scenarios.