Problem 44 - A Complete Resource Book in Chemistry for JEE Main

Question

If a gas expands at constant temperature (1) the pressure decreases (2) the kinetic energy of the molecules remains the same (3) the kinetic energy of the molecules decreases (4) the number of molecules of the gas increase (a) 1,2 (b) \(1,2,3\) (c) \(1,2,4\) (d) 2,3

Step-by-Step Solution

Answer

The correct answer is (a) 1,2.

Step 1: Analyzing the process

The exercise involves an expansion of a gas at constant temperature, which represents an isothermal process. In such a process, temperature is constant while the gas expands.

Step 2: Applying the Ideal Gas Law

For an isothermal process, the Ideal Gas Law can be stated as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature. As \( T \) is constant, \( P \) and \( V \) are inversely related, meaning if volume \( V \) increases, pressure \( P \) decreases.

Step 3: Analyzing Kinetic Energy

The kinetic energy of gas molecules is directly related to the temperature of the gas. Since the temperature remains constant during the isothermal expansion, the average kinetic energy of the gas molecules also remains unchanged.

Step 4: Evaluating the Number of Molecules

The number of molecules of gas does not increase as a result of an isothermal expansion, only the volume increases while the number of molecules, \( n \), does not change.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the state of an ideal gas. This law combines several gas laws, including Boyle's, Charles', and Avogadro's, into a single equation: \[ PV = nRT \] Here:

• \( P \) stands for the pressure of the gas.
• \( V \) is the volume it occupies.
• \( n \) represents the number of moles of the gas.
• \( R \) is the universal gas constant.
• \( T \) is the temperature measured in Kelvin.

In an isothermal process, such as the one described in the exercise, the temperature \( T \) remains constant. This means that if the gas expands (increasing volume \( V \)), then the pressure \( P \) must decrease to maintain the equality. Understanding this principle helps explain why the pressure decreases during an isothermal expansion of a gas.

Kinetic Energy

Kinetic energy in the context of gases refers to the energy that gas molecules possess due to their motion. The average kinetic energy of molecules in a gas is directly proportional to the temperature of the gas:\[ E_k = \frac{3}{2}kT \]where:

• \( E_k \) is the average kinetic energy per molecule.
• \( k \) is the Boltzmann constant.
• \( T \) is the absolute temperature of the gas.

During an isothermal expansion, since the temperature remains constant, the average kinetic energy of the molecules remains the same. This concept is crucial because it differentiates isothermal processes from others, such as adiabatic processes, where temperature and kinetic energy would change.

Gas Expansion

Gas expansion refers to the increase in volume a gas experiences under certain conditions. In the scenario of isothermal expansion, the gas spreads out to fill a larger volume while maintaining constant temperature. This expansion is characterized by a reduction in pressure due to the increase in volume.

• During an isothermal process, the expansion does not affect the kinetic energy or the number of molecules.
• Understanding gas expansion helps in various real-world applications, such as engines and refrigeration.

Because temperature is a constant during isothermal expansion, the process is ideal for studying how volume and pressure interact without the complication of changing temperatures.

Pressure-Volume Relationship

The pressure-volume relationship in gases is a core concept in understanding gas behavior. This relationship is evident through Boyle's Law, which states that for a given amount of gas at constant temperature, the pressure is inversely proportional to the volume:\[ P \times V = \, \text{constant at constant } T \]As a gas expands isothermally, increasing its volume \( V \), its pressure \( P \) decreases. This inverse relationship is depicted as a hyperbolic curve on a PV graph.This principle is fundamental in examining processes involving gases, making it essential to grasp how changes in volume affect pressure in real-life scenarios and scientific studies. It helps engineers and scientists predict outcomes of gas manipulations in varied environments. Keeping this relationship in mind is key to solving problems involving gas dynamics.