Question

(a) Calculate the total rotationalkinetic energy of the molecules in of a diatomic gas at. (b) Calculate the moment of inertia of an oxygen molecule for rotation about either the or shown in Fig. 18.18b. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of . The molar mass of oxygen atomsis (c) Find the rms angular velocity of rotation of an oxygen molecule about either they- or z-axis shown in Fig. 18.18b. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery ?

Short answer

(a)Total totalrotationalkinetic energy is

(b)The moment of inertia of oxygen molecule is

(c) RMS angular velocity of the oxygen molecule is

Step-by-step solution

Step 1:Determine the total rotational kinetic energy of the molecules in  of a diatomic gas at.

The kinetic energy of a rotating object is analogous tolinear kinetic energyand can be expressed in terms of themoment of inertiaandangular velocity.

Solve for totalrotationalkinetic energy

Therefore total total rotational kinetic energy is

Step 2:Determine moment of inertia of oxygen molecule 

Moment of inertia can be defined as a quantity that expresses a body’s tendency to resist angular acceleration, which is equal to the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation

solve for moment of inertia

Therefore , the moment of inertia of oxygen molecule is

Step 3:Determine the RMS angular velocity of the molecule 

Solve for angular velocity

Therefore ,the RMS angular velocity of the oxygen molecule is