(a) Calculate the total rotational kinetic energy of the...

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 $\mathrm{K}$ . (b) Calculate the moment of inertia of an oxygen molecule $\left(\mathrm{O}_{2}\right)$ for rotation about either the $y$ - or $z$ -axis shown in Fig. 18.18 . Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of $1.21 \times 10^{-10} \mathrm{m}$ . The molar mass of oxygen atoms is $16.0 \mathrm{g} / \mathrm{mol} .$ (c) Find the rms angular velocity of rotation of an oxygen molecule about either the $y$ - or $z$ -axis shown in Fig. 18.15. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery $(10,000 \mathrm{rev} / \mathrm{min}) ?$

Key Concepts

Rotational Kinetic Energy

This is the energy associated with an object's rotation, given by the formula (1/2)I?², where I is the moment of inertia and ? is the angular velocity. In the context of molecules, each rotational degree of freedom contributes an average energy of (1/2)kT per molecule, according to the equipartition theorem, which allows us to calculate the total rotational energy for a collection of molecules at a given temperature.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation about a given axis. For a diatomic molecule modeled as two point masses, it depends on the masses of the atoms and the square of half the distance between them if rotating about an axis perpendicular to the bond. It plays a crucial role in determining the rotational kinetic energy and the angular velocity of the molecule.

Root Mean Square (RMS) Angular Velocity

The RMS angular velocity is a statistical measure that represents the typical angular speed of molecules in a gas at a given temperature. It is derived from equating the rotational kinetic energy per degree of freedom (derived from the equipartition theorem) with (1/2)I?², thus linking the thermal energy with the molecular rotation.

Equipartition Theorem

The equipartition theorem is a principle from statistical mechanics stating that at thermal equilibrium each quadratic degree of freedom contributes (1/2)kT of energy per molecule. For diatomic gases, since there are two rotational degrees of freedom (perpendicular to the axis of the bond), this theorem helps to distribute the thermal energy among the possible modes of motion.

Transcript

00:01 For this problem on the topic of thermal properties of matter, we wish to calculate the total rotational kinetic energy of one mole of the diatomic gas at a given temperature.

00:10 We want to calculate as well the moment of inertia of an oxygen molecule, provided that we are given the distance that separates the oxygen atoms.

00:22 We finally want to find the rms angular velocity of rotation of the oxygen molecule about either the y or x axis, as shown in the diagram.

00:30 And we want to compare this to a piece of rapidly rotating machinery.

00:36 So firstly, we know that the two degrees of freedom associated with the rotation for a diatomic molecule account for two -fifths of the total kinetic energy.

00:47 So the rotational kinetic energy, k, is equal to n r t.

00:56 And if we substitute our values into this, we know that we're looking for a mole of the diatomic molecule.

01:03 Times the universal gas constant of 8 .3145 and that's joules per mole kelvin times a temperature of 300 kelvin so the average kinetic energy due to the rotations is 2 .49 times 10 to the 3 joules so that's the portion of the kinetic energy that comes due to the rotation of the diatomic molecule.

01:46 Now secondly we want to calculate the moment of inertia for the diatomic oxygen molecule.

01:52 Now we know the moment of inertia i is equal to 2m times l over 2 all square, where l is the distance that separates the two oxygen atoms.

02:07 So since we are given the mass we can calculate this moment of inertia...

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