b. A nitrogen molecule consists of two nitrogen atoms separated by 0.11 nm, the bond length. Treat the molecule as a rotating dumbbell and find the rms angular velocity at this temperature of a nitrogen molecule around the z-axis, as shown in Figure 20.10.

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Hey, everyone. Let's go through this problem. The hydrogen diatomic molecule modeled as a rigid molecule revolves around a vertical rotational axis as shown in the figure, the separation distance between the two hydrogen atoms is 20. angstroms, calculate the angular frequency at 350 Kelvins of the hydrogen molecule revolving around the vertical axis. Option A 3.2 multiplied by 10 to the power of four radiant per second. Option B 5.4 multiplied by 10 to the power of five radiant per second. Option C 4.6 multiplied by 10 to the power of radiant per second. And option D 8.1 multiplied by 10 to the power of radiant per second. Since we're looking for the angular frequency of the molecule, we can relate that to the molecule's angular its rotational kinetic energy. And we can find the rotational kinetic energy using the thermodynamic properties described to us about the molecule. In the problem. Recall that according to the equi partition theorem, the energy of a molecule can be described by saying that for every degree of freedom that a molecule has each degree of freedom contributes one half multiplied by the boltzmann constant multiplied by the temperature of the molecule. So this entire term multiplied by the number of degrees of freedom for the molecule is the energy of the molecule that said. However, in this particular case, we're only interested in the rotational kinetic energy. And since we're explicitly told in the problem that the molecule is diatomic recall that for a diatomic molecule, there are two rotational degrees of freedom. So we're going to take this formula for the rotational energy and multiply it by two. So that means that the rotational kinetic energy of the molecule is equal to just K T the Boltzmann constant multiplied by the temperature because the two cancels that with the one half. So let's put this into a calculator to find the total rotational energy. This is the Boltzmann constant, which is 1.38 multiplied by 10 to the power of negative 23 Jews per Kelvin multiplied by the temperature 350 Kelvins put that into a calculator. And the rotational kinetic energy is about 4. multiplied by 10, raised the power of negative 21 Jews in order to find a relationship between this quantity of energy and the angular frequency to recall from classical mechanics. But the rotational kinetic energy can also be written as one half multiplied by the moment of inertia of the rotating body multiplied by the square of the angular frequency. This gives us the relationship we need. So what we need to do is solve this equation algebraically for omega for the angular velocity. So algebraically, we can see by dividing both sides of the equation by one half, I, we can see that Omega squared is equal to two multiplied by the rotational energy divided by the moment of inertia. So logically, by taking the square root of both sides, we can see that the angular velocity on its own is equal to the square root of the rotational energy divided by the moment of inertia. But now we'll need to actually find the moment of inertia for the molecule. Fortunately, for us with the way this problem in the molecule is described finding the moment of inertia is actually fairly simple because recall from classical mechanics that the moment of inertia of one single point particle with some mass is equal to the mass of that particle multiplied by the square of the radial distance between the center of where the rotation is happening. And where the point we're analyzing is. Well, for us, there are only two point particles we're looking at where the two hydrogen atoms are. So the moment of inertia of this molecule is simply going to be twice that because it's just the moment of inertia of one particle multiplied by two. Let's quantify the moment of inertia. Now, so it's equal to two multiplied by the mass of just one of the hydrogen atoms. And the hydrogen atom has a mass of about 1. atomic mass units. And we're gonna want to convert this into kilograms by multiplying it by 1.66, multiplied by 10 of the power of negative 27 kg. That's the mass exchange rate for one atomic mass unit. Then we multiply this by the square of the radial distance between a hydrogen atom and the center. And we can see from the diagram that the radial distance is just going to be half of the total distance between the two atoms. So one half of 10.74 angstroms. So 0.74 and angst drums to convert from angst drums to meters, that's multiplied by 10 to the power of negative meters. We divide by two to get the radial distance and square that whole term. And if we put that into a calculator, then we find the moment of inertia for the molecule is 4.58 multiplied by to the power of negative kilogram meter squared. All that's left for us to do now is plug the values we found into the angular velocity equation we discussed earlier. So that's the square root of two multiplied by the total energy, the rotational energy that is of the molecule multiplied by 4.83 multiplied by 10 of the power of negative 21 Jews. And this is being divided by the moment of inertia, which we just found to be 4.58 multiplied by 10 to the power of a negative 48 kg meter squared. If we put all of this into a calculator, then we find an annual velocity of approximately 4.6 multiplied by 10 to the power of 13 radians per second. So this is our answer to this problem. And if we look at our multiple choice options, this agrees with option C. So option C is the correct answer to this problem. And that is it for this video. I hope this video helped you out. If you believe you need more practice, please check out some of our other tutoring videos which will give you more experience with these types of problems, but that's all for now. And I hope you all have a lovely day. Bye bye.

b. A nitrogen molecule consists of two nitrogen atoms separated by 0.11 nm, the bond length. Treat the molecule as a rotating dumbbell and find the rms angular velocity at this temperature of a nitrogen molecule around the z-axis, as shown in Figure 20.10.

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Key Concepts

Bond Length

RMS Angular Velocity

Rotational Motion