Problem 138 - Master Resource Book in JEE Main Physics

Question

\(A B C D E F G H\) is hollow cube made of an insulator. Face \(A B C D\) has positive charge on it. Inside the cube, we have ionized hydrogen. The usual kinetic theory expression for pressure [NCERT Exemplar] (a) will be valid (b) will not be valid since the ions would experience forces orger than due to collisions with the walls (c) will not be valid since collisions with walls would not be elastic (d) will not be valid because isotropy is lost

Step-by-Step Solution

Answer

The pressure expression of kinetic theory is not valid because isotropy is lost.

Step 1: Understanding the Situation

In this problem, we're dealing with a hollow cube made of an insulating material which is charged on one face. Inside this cube, there is ionized hydrogen. The key point is that the charge on the face will exert an electrostatic force on the ions inside.

Step 2: Applying Kinetic Theory

The kinetic theory of gases assumes certain ideal conditions: no interaction between particles except during elastic collisions, and isotropy in all directions. These conditions may not hold in this scenario due to the charged face influencing the motion of ions.

Step 3: Assessing Forces

The electrostatic force due to the charged face on the ions will be significantly stronger than the forces arising from the collisions of ions with the walls. This affects the motion of the ions and thus the kinetic theory assumptions.

Step 4: Analyzing Collision Nature

Kinetic theory assumes perfect elasticity in collisions, but the influence of the electric field may change the energy distribution during these collisions, potentially making them inelastic.

Step 5: Evaluating Isotropy

Due to the presence of the charge on one face, the system may lose isotropy. The electric field causes the movement of ions to be biased towards or away from the charged face, violating a fundamental assumption of isotropy in kinetic theory.

Step 6: Conclusion Based on Analysis

Considering these analyses, the usual kinetic theory expression for pressure will not be valid, primarily because isotropy is compromised and the forces due to the electric charge overpower the usual collisional forces considered in kinetic theory.

Key Concepts

Electrostatic Force

The electrostatic force is a crucial concept when discussing interactions between charged particles. In our scenario, this force is significant due to the charged face of the cube housing ionized hydrogen. This face carries a positive charge, which means it will exert a force on any charged particle, like the ions inside the cube. This force is described by Coulomb's law, which states that the force between two charges is proportional to the product of the amounts of the charges and inversely proportional to the square of the distance between them.

In simpler terms, the closer the ions are to the charged face, the stronger the force they experience. This force affects the motion and behavior of the ionized hydrogen within the cube, making it a critical factor in this problem. Compared to forces from collisions with the cube walls, the electrostatic force is likely to be much stronger and more influential, thereby impacting the assumptions of the kinetic theory of gases.

Isotropic Conditions

Isotropic conditions are a foundational assumption in the kinetic theory of gases. "Isotropic" means that the properties are the same in all directions. This implies that particles within a gas move randomly and uniformly in all directions, with no preferred path or bias.

In the context of our exercise, however, this assumption falls apart due to the charged face of the cube. The electric field generated by this charge influences the movement of the ionized hydrogen, creating a directional bias. Particles are no longer free to move equally in all directions; instead, they may drift towards or away from the charged face.

• This results in a loss of isotropy because the external electrostatic force disrupts the random and equal distribution of particle velocities.
• The lack of isotropy invalidates the usual kinetic theory predictions about gas pressure and behavior within this system.

Elastic Collisions

Elastic collisions are another assumption central to the kinetic theory of gases. In an elastic collision, the total kinetic energy and momentum of the colliding particles are conserved. This means no energy is lost to other forms like sound or heat during a collision.

In the scenario of the hollow cube, the presence of an electric field can alter these collisions' characteristics. The charged face can influence the velocities and energy distribution among ions, potentially turning some collisions inelastic.

• When collisions become inelastic, some energy is lost to internal energy or other forms, altering the expected pressure and movement described by kinetic theory.
• Therefore, the assumption of elastic collisions doesn't hold firmly here, affecting how predictably the gas behaves under kinetic theory assumptions.

Ideal Gas Assumptions

The kinetic theory of gases relies heavily on several idealized assumptions:

• Particles are in constant, random motion, exhibiting no attractions or repulsions except during collisions.
• Collisions are entirely elastic.
• The system is isotropic, with no external forces causing directional bias.

However, these assumptions break down in our charged cube scenario. The electrostatic force from the charged face creates significant interactions that cannot be ignored.

• These forces cause deviations from random motion and isotropy, making the usual kinetic energy and velocity distributions unreliable.
• The inelastic nature of collisions further skews predictable outcomes based on ideal assumptions.

Since these ideal gas assumptions don't hold under the influence of strong external forces like electrostatic forces, the typical kinetic theory expressions for gas pressure and behavior won't apply accurately.