A spherical gas chamber of radius R filled with a light gas, in which two spherical balls A and B of radius Ra and Rb simultaneously are suspended at the distance d, we have to find formula to calculate-

1. Pressure difference of gas molecules on the surfaces of balls A and B due to their shadow effect?

2. Pressure generated by gas molecules on the internal surfaces of both balls due to multiple collisions of gas molecules when distance d is short?

(Assuming mean free path of gas particles and radius R of gas chamber tending towards infinity and collision of gas particle with balls is totally elastic, N is the number of gas molecules striking per unit area of balls and m is mass of a gas molecule)

To tackle the problem of pressure differences around two suspended spherical balls in a gas chamber, we need to consider the effects of gas molecules colliding with the surfaces of the balls and how their sizes and positions influence the pressure experienced by each ball. Let's break this down into two parts as per your question.

1. Pressure Difference Due to Shadow Effect

The shadow effect arises because the two balls obstruct the flow of gas molecules, leading to a difference in pressure on their surfaces. When gas molecules collide with the balls, they exert force on their surfaces. The pressure difference can be derived from the concept of molecular flux and the effective area of each ball.

Calculating the Pressure Difference

Let’s denote:

• R_a = radius of ball A
• R_b = radius of ball B
• d = distance between the centers of the two balls
• P_a = pressure on ball A
• P_b = pressure on ball B

The pressure exerted by gas molecules on a surface can be expressed as:

P = N * m * v_avg

Where:

• N = number of gas molecules striking per unit area
• m = mass of a gas molecule
• v_avg = average velocity of gas molecules

For each ball, the effective area that gas molecules can strike is influenced by their size. The pressure difference can be approximated as:

ΔP = P_a - P_b = (N_a - N_b) * m * v_avg

Where N_a and N_b are the number of molecules striking balls A and B, respectively. The shadow effect means that the effective area for ball A is reduced by the presence of ball B and vice versa. This can be modeled as:

N_a = N * (1 - (R_b/d)²)

N_b = N * (1 - (R_a/d)²)

Substituting these into the pressure difference equation gives:

ΔP = (N * (1 - (R_b/d)²) - N * (1 - (R_a/d)²)) * m * v_avg

2. Pressure Generated by Gas Molecules on Internal Surfaces

When considering the pressure generated by gas molecules on the internal surfaces of both balls, we need to account for the mean free path of the gas particles and the elastic collisions occurring within the confined space of the balls.

Understanding Internal Pressure

The internal pressure can be derived from the kinetic theory of gases. The pressure inside a spherical ball due to gas molecules can be expressed as:

P_internal = (2/3) * N * m * v_avg / V

Where V is the volume of the ball. For a sphere, the volume is given by:

V = (4/3) * π * R³

Thus, the internal pressure for each ball becomes:

P_A = (2/3) * N_A * m * v_avg / ((4/3) * π * R_a³)

P_B = (2/3) * N_B * m * v_avg / ((4/3) * π * R_b³)

As the distance d becomes small, the number of gas molecules striking the internal surfaces increases, leading to higher internal pressures. The mean free path is assumed to be small enough that collisions are frequent, and thus the pressure can be approximated as:

P_internal = (N * m * v_avg) / (4 * π * R²)

This equation shows that the internal pressure is directly proportional to the number of gas molecules and their average velocity, while inversely proportional to the surface area of the balls.

In summary, the pressure difference due to the shadow effect and the internal pressure generated by gas molecules can be calculated using the principles of kinetic theory and the geometry of the system. These concepts illustrate how the interactions between gas molecules and suspended objects can lead to significant variations in pressure within a gas chamber.