(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 K. (b) Calculate the moment of inertia of an oxygen molecule ($O_2$) for rotation about either the $y$- or $z$-axis shown in Fig. 18.18b. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of $1.21 \times 10^{-10}$ m. The molar mass of oxygen $atoms$ is 16.0 g/mol. (c) Find the rms angular velocity of rotation of an oxygen molecule about either the $y$- 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 (10,000 rev/min)?

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a given axis. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation. For simple systems like diatomic molecules modeled as two point masses, the moment of inertia can be calculated by multiplying each mass by the square of its distance from the axis of rotation.

Diatomic Molecule Model

A diatomic molecule can be modeled as two point masses connected by a bond, with the masses separated by a fixed distance. This simplified model is used to approximate the rotational dynamics of the molecule by considering the two atoms as masses contributing to the overall moment of inertia about axes perpendicular to the internuclear axis. This approach provides insights into molecular rotational energy levels and spectra.

Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with the rotation of an object. It depends on the object's moment of inertia and its angular velocity, analogous to how translational kinetic energy depends on mass and linear velocity. In systems like molecules, this energy contributes to the total internal energy and plays an important role in understanding thermodynamic and dynamic behavior at the microscopic scale.

Equipartition Theorem

The equipartition theorem is a fundamental principle in statistical mechanics that states that energy is equally distributed among all accessible quadratic degrees of freedom in a system at thermal equilibrium. For rotational motion, each rotational degree of freedom contributes 1/2 kT of energy per molecule. This concept links the microscopic behaviors of molecules to macroscopic thermodynamic properties like temperature.

Root-Mean-Square (RMS) Angular Velocity

The root-mean-square (RMS) angular velocity is a statistical measure of the magnitude of the angular velocities in a collection of molecules or systems. It provides a way of averaging over the various possible angular velocities to give a representative value, which is particularly useful in thermally agitated systems where the individual motions vary widely.

Transcript

00:01 So we need to find the total rotational kinetic energy of a diatomic gas, in this case, oxygen.

00:13 And then we are going to an oxygen gas, a o2 molecule.

00:17 And then we need to find the moment of inertia on the y or z axis for this molecule.

00:25 And then we need to find the root mean squared angular velocity.

00:30 For this rotational motion.

00:34 So it's a bit difficult in the sense that we need to use a bit more, we need to use a bit more formulas from other chapters.

00:46 So let's first write our down our given.

00:50 So n equals one mole.

00:52 So we have one mole.

00:53 And then it's important to mention that it's a diatomic gas.

01:00 And then we have t equals 300 kelvin.

01:06 And then we have a definition for rotational kinetic energy being 1 over 2 times the moment of inertia, times the angular velocity squared.

01:20 And we know that the moment of inertia for a diatomic molecule would be 2 times the mass times l over 2 squared, where l is the distance equals distance between two atoms in a molecule, and then m would equal the mass of one atom.

02:00 So again, one atom, not one molecule.

02:03 So essentially, we would have to, let's solve for this really quickly mean just say this is going to be equal to ml squared over 4.

02:17 And at this point, we can say, okay, let's start off with a diatomic molecule.

02:23 For a diatomic molecule, let's draw it out here.

02:36 If these are your axes, if these are your axes, maybe right here, right here, here.

02:53 If these are your axes, you can only rotate two ways.

02:59 You can rotate on the z axis, and you can, if this was the z, and then this was the, we can call this, we can call this a y, and then we can call this the x.

03:12 You can rotate on the x axes, and you can rotate on the z axes, rather, i think the actual question said z or y.

03:23 So, in fact, according to the question, this would actually be a y, and this would be a lot of so according to the question, we can rotate on the z and the y axes because that would change the actual orientation.

03:42 But if you were to rotate a diatomic molecule on the axial length, it wouldn't, it would be as if you're rotating a sphere on the, on that, on any axes.

03:56 So it's not as if, so we have two degrees of freedom of, two degrees of freedom for rotational motion.

04:02 And one degree is going to be, it should be three, but one degree actually doesn't change the look of the atom whatsoever.

04:13 So rotating it on the x -axis wouldn't really change how the diatomic molecule looks.

04:20 So if you can imagine rotating it on the z -axis because that would actually create a different, you know, a different orientation rotating on the y.

04:33 Axis would create a different orientation.

04:35 However, on the z axis, straight through the middle, it wouldn't change the orientation.

04:43 So if we were to say this, two degrees of freedom for the rotational kinetic energy would be equal to the degrees of freedom over two times nrt, or it would be equal to 2 over 2, so it simply be equal to nrt, and we have the number of moles, so 1 times 8 .3, 14 times t of 300 kelvin.

05:22 So this is going to be equal to 2 .49 times 10 to the 3rd joules.

05:34 10 to the 3rd joules.

05:35 So at this point, this would be our answer for part a.

05:41 This would be the total rotational motion associated with a diatomic molecule at 300 kelvin for one mole of this gas.

05:52 B, this is asking us to find the imamative inertia for this molecule.

05:57 So we need to find the mass of one molecule.

06:00 So it would be m over avagagos number.

06:05 This would be equal to 0 .032 kilograms per mole.

06:15 And i apologize, this is going to be molecule.

06:23 And then we can divide this by 6 .0 to 3 times 10 to the 23rd...