Question

Molecules in a spherical container make wall collisions in a great circle plane of the sphere. All paths traveled between collisions are equal in length. Relate the path length to the angle and the sphere radius r.

a) Path length is directly proportional to the angle and inversely proportional to the square of the sphere radius
b) Path length is directly proportional to the sphere radius and inversely proportional to the angle
c) Path length is directly proportional to the square of the sphere radius and inversely proportional to the angle
d) Path length is directly proportional to the angle and the sphere radius

Answer 1

The path length traveled by molecules between collisions in a spherical container is directly proportional to both the sphere radius and the angle of rotation, as represented by the relationship s = rα.

Step-by-step explanation:

The student's question pertains to the relationship between the path length traveled by molecules between collisions in a spherical container, the angle of rotation, and the sphere's radius r. The relationship between these variables is derived from the concept of arc length in a great circle of the sphere, which is a section of the circumference of the sphere.

If we consider the arc length s to be the path length, then the fundamental relationship between the arc length, the angle α (in radians), and the radius r of the circle is given by:

s = rα

This equation shows that the path length (s) is directly proportional to both the radius r and the angle α. Therefore, as the radius or the angle increases, so does the arc length, meaning that the path length traveled by molecules between collisions is directly proportional to the sphere radius and the angle of rotation.