(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 $\mathrm{b}$ . 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. 18.18 $\mathrm{b}$ . How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery $(10,000$ rev/min $)$ ?

(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 $\mathrm{b}$ . 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. 18.18 $\mathrm{b}$ . How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery $(10,000$ rev/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 Okay, so we know the rotational kinetic energy in this case can be equal to nrt.

00:04 N is the mole.

00:05 R is the universal gas constant, which is 8 .314 g per mole times kelvin, and t is the temperature, which is given as 300 kelvin.

00:13 And when the mole is one more, so therefore if you plug in battery equation, roughly rotational kinetic energy is equal to 1 .00 mole times 8 .314 jiu per mole times kelvin and n times 300 kelvin and this will give us the rotational kinetic energy is about 2 .49 times 10 to the power 3 g.

00:48 For the next question, while we know the moment of inertia, which is the i here can be equal to 2m times d over 2 to the power 2.

00:55 Okay? so m here is the mass of each addens and d here is the distance between two atoms in one molecule.

01:07 So if we do some arrangement here, eventually you have i is equal to m d squared over 2.

01:11 We know m here, which is a mass for each addon, which is capital m over an a.

01:16 We know the motor mass for the oxygen is 16 gram per more, and avogadal number is 6 .02 times 10, 10 to power, 23 per more.

01:24 So therefore, eventually, we'll have the individual mass for each oxygen addin is about 2 .66 times 10 to 0 .23 grams.

01:31 If we convert to kilograms, it's about 2 .66 1010 to power negative 26 kilograms.

01:38 And when we know d, which is the distance between 2 adams, is given as 1 .21 times 10 to 90 -10 meter.

01:44 So therefore, if we plug in back to the equation, we'll have the minimum of inertia.

01:50 It's actually equal to 2 .66 times 10 to the power of negative -26 kilogram, and then times 1 .21 times 10 to the power of negative 10 meter, then to the power 2, over 2.

02:16 And this will give us the moment inertia is equal to 1 .95 times 10 to the power of negative 46 kilograms times meter square...

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