Thermal Properties - Energy and velocity of molecules

[forestmine]

Thermal Properties -- Energy and velocity of molecules

Hello all,

Homework Statement

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300K.

(b) Calculate the moment of inertia of an oxygen molecule. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of 1.21 x 10^-10 m. The molar mass of oxygen atoms is 16g/mol.

(c) Find the rms angular velocity of rotation of an oxygen molecule. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery (10,000 rev/min)?

Homework Equations

KE = 1/2Iω^=3/2kt
v(rms)=√(3kt/m)
Moment of inertia for this particle case...?

The Attempt at a Solution

I'm honestly not at all sure how to go about this. My initial thought is to use KE=1/2Iω^2 for the first portion, but I do not see how to consider temperature given that equation alone. For part B, I'm not sure which moment of inertia equation is correct for the two points rotating about an axis...I'm thinking 1/12mL^2, but I'm really not sure.

Any help or direction at all would be greatly appreciated. Thank you!

 

[anathema]

Hello!

I am happy to help you with this problem. To start, let's review some basic concepts related to thermal properties and energy of molecules.

Temperature is a measure of the average kinetic energy of the molecules in a substance. In this case, we are dealing with a diatomic gas, which means it is made up of two atoms (in this case, oxygen). The kinetic energy of a molecule can be divided into three types: translational, rotational, and vibrational. Translational kinetic energy is the energy associated with the linear motion of the molecule. Rotational kinetic energy is the energy associated with the rotation of the molecule around its center of mass. And vibrational kinetic energy is the energy associated with the vibration of the atoms within the molecule.

In order to calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300K, we need to use the equation KE = 1/2Iω^2, where I is the moment of inertia and ω is the angular velocity. The total rotational kinetic energy will be the sum of the individual rotational kinetic energies of each molecule in the gas. We can calculate the moment of inertia using the equation I = mr^2, where m is the mass of the molecule and r is the distance between the two atoms. We know that the molar mass of oxygen atoms is 16g/mol, so the mass of one oxygen molecule (which has two oxygen atoms) would be 32g/mol. We also know that the distance between the two oxygen atoms is 1.21 x 10^-10 m. Plugging these values into the equation, we get I = (32g/mol)(1.21 x 10^-10 m)^2 = 4.90 x 10^-46 kg*m^2.

Now, we need to calculate the angular velocity (ω) of the molecules. We can use the equation v(rms) = √(3kt/m), where v(rms) is the root mean square velocity, k is the Boltzmann constant (1.38 x 10^-23 J/K), t is the temperature in Kelvin, and m is the mass of the molecule. Plugging in the values, we get v(rms) = √(3(1.38 x 10^-23 J/K)(300K)/(32g/mol)) = 5