Moment of Inertia: Ignoring Small Bond Axis Molecule

[gracy]

Why the moment of inertia of a molecule about the bond axis is small enough to ignore ?

 

[Simon Bridge]

Try calculating it and see.

 

[gracy]

Try calculating it and see.

I really don't have any idea.Please guide me.

 

[Bystander]

Where is the mass of an atom located?

 

[gracy]

Where is the mass of an atom located?

throughout the atom.But being more technical i think where center of mass is present.

 

[Simon Bridge]

How do you normally calculate the moment of inertia?
What would the distribution of mass in a diatomic molecule look like - roughly?

 

[Bystander]

throughout the atom

"?" Back up. Describe the structure of an atom.

 

[gracy]

"?" Back up. Describe the structure of an atom.

sorry i think the nucleus contain all mass as electrons have negligible mass.

 

[Bystander]

Better. What's the diameter of a typical nucleus, or has that been covered for you at all?

 

[gracy]

How do you normally calculate the moment of inertia?
What would the distribution of mass in a diatomic molecule look like - roughly?

I normally calculate moment of inertia as total mass multiplied by distance square.

 

[gracy]

Better. What's the diameter of a typical nucleus, or has that been covered for you at all?

about 1.75 fm.

 

[Bystander]

Sounds about right. What's nuclear mass x 1/2 of that squared?

 

[gracy]

What's nuclear mass x 1/2 of that squared?

Sorry I didn't understand.Nuclear mass would depend on atom of which element we are taking.

1/2 of that squared?

squared of what?

 

[gracy]

I normally calculate moment of inertia as total mass multiplied by Distance between the axis and rotation in m..

 

[Bystander]

Sorry I didn't understand.Nuclear mass would depend on atom of which element we are taking.

Pick an atom.

squared of what?

"What?" The nuclear diameter. That's an end point of a bond axis.

 

[gracy]

That's an end point of a bond axis.

Sorry i am talking about a molecule.Here x-axis is bond axis.

 

[Bystander]

No, that is one of three rotational axes directed at right angles to one another through the center of mass of the molecule; the bond axis is the line connecting the nuclei of the two atoms, and is the one axis of the three that you're interested in for this problem.

 

[Vagn]

No, that is one of three rotational axes directed at right angles to one another through the center of mass of the molecule; the bond axis is the line connecting the nuclei of the two atoms, and is the one axis of the three that you're interested in for this problem.

Bystander is correct, the axis you have as your x-axis is one with a non-negligible moment of inertia. There would also be one perpendicular to this and the axis along the bond, which also has a non-negligible moment of inertia, this is why diatomic gases have two rotational degrees of freedom. The bond axis is that which joins the two nuclei.

 

[gracy]

[QUOTE="Bystander, post: 4940685, member: 50"the bond axis is the line connecting the nuclei of the two atoms,

Now is x-axis bond axis?

 

[Vagn]

[QUOTE="Bystander, post: 4940685, member: 50"the bond axis is the line connecting the nuclei of the two atoms,
View attachment 76453
Now is x-axis bond axis?

No, your black line (the bond) is (lies along) the bond axis.

 

[gracy]

No, your black line (the bond) is (lies along) the bond axis.

That's what i mean. Is black line bond axis?

 

[Vagn]

That's what i mean. Is black line bond axis?

The black line joining the two blue atoms is the bond axis. Your diagram is confusing as the arrow suggests your x-axis is vertical and perpendicular to the bond.

 

[gracy]

No, your black line (the bond) is (lies along) the bond axis.

 

[gracy]

The black line joining the two blue atoms is the bond axis. Your diagram is confusing as the arrow suggests your x-axis is vertical and perpendicular to the bond.

Right?

 

[Vagn]

View attachment 76455
Right?

That's better.

 

[gracy]

That's better.

Now how to proceed further in order to get the answer .

 

[Bystander]

What's the maximum distance of the nuclear mass from that axis?

 

[gracy]

What's the maximum distance of the nuclear mass from that axis?

here (in the image i have given)there are two atoms( after all i am referring to diatomic molecule)nuclear mass of which one?

 

[Bystander]

Both of them. How far are they from the axis you've drawn?

 

[gracy]

Both of them. How far are they from the axis you've drawn?

I think answer would be ~nearly zero.

 

[Bystander]

Yes. This is the point of asking you what nuclear dimensions are; so, calculating moments of inertia for the three rotational axes of a molecule about its center of mass, you have two that square something of the same order of magnitude as the bond length (not half in general, because not all diatomic molecules are homonuclear), and a third moment that squares a distance the order of nuclear dimensions.

 

[gracy]

This is the point of asking you what nuclear dimensions are; so, calculating moments of inertia for the three rotational axes of a molecule about its center of mass, you have two that square something of the same order of magnitude as the bond length (not half in general, because not all diatomic molecules are homonuclear), and a third moment that squares a distance the order of nuclear dimensions.

Sorry but i am not getting this post.

 

[Bystander]

Three questions: what size are bond lengths? What is typical nuclear diameter (radius)? What is the ratio of the squares of those two dimensions?

 

[gracy]

bond length is the average distance between nuclei of two bonded atoms in a molecule. why are you asking for bond lengths?as there is only one ,right?[/QUOTE]
and why you have written radius inside the bracket after word diameter as these two are totally different.

 

[Bystander]

You've got finals or midterms coming up, so I'll give you a break --- squaring bond lengths gives you something of the order of 10^-20m² and squaring nuclear diameter or radius gives you something of the order of 10^-30m². Multiplying those values time nuclear masses gives you moments of inertia. The moment of inertia about a bond axis is on the order of 10^-10 that of the moments of inertia for rotations about axes perpendicular to the bond axis.

 

[gracy]

You've got finals or midterms coming up, so I'll give you a break --- squaring bond lengths gives you something of the order of 10^-20m² and squaring nuclear diameter or radius gives you something of the order of 10^-30m². Multiplying those values time nuclear masses gives you moments of inertia. The moment of inertia about a bond axis is on the order of 10^-10 that of the moments of inertia for rotations about axes perpendicular to the bond axis.

Please can you explain how you calculate moment of inertia.I am having very hard time understanding it .what exactly is bond length (bond axis)here (in the image given below)am i correct?

 

[gracy]

nuclear diameter or radius

how these two are same?

 

[Bystander]

Please can you explain how you calculate moment of inertia.

I normally calculate moment of inertia as total mass multiplied by distance square.

You've already said how to calculate the moment of inertia.

how these two are same?

They are the same order of magnitude. One is half the other, diameter is twice the radius.
Bond length is the internuclear distance.

 

[gracy]

You've already said how to calculate the moment of inertia.I normally calculate moment of inertia as total mass multiplied by distance square.

I am not sure which distance square?

 

[Bystander]

The distance from the axis of rotation to the increment of mass.

 

[K41]

I am not sure which distance square?

Its the distance perpendicular from whatever axis you are taking it from. You can have moments of Inertia about the y-axis and about the x axis.

Iyy = summation of m_i*x_i² (its in x direction because x is perpendicular to the y axis).
Ixx = summation of m_i*y_i² (its in y direction because y is perpendicular to the x axis).

Of course, you could have moment of inertia about any axis and that is where things become more complicated mathematically.

 

[gracy]

please have a look at this video from time 3:15 to 3:30.Why he took moment of inertia about x-axis nearly zero?

 

[gracy]

squaring bond lengths gives you something of the order of 10-20m2

How you calculate bond lengths without any information given?

 

[Bystander]

Bond lengths between atoms are in the neighborhood of 10^-10 m. For a specific compound you can measure bond length using X-ray diffraction, or calculate a bond length from spectroscopic data. All bond lengths fall into a range from ~1 x 10^-10 to 2 x 10^-10m.

 

[gracy]

according to you Moment of inertia=total nuclear mass multiplied by The distance from the axis of rotation to the increment of mass.And according to your post no 35 you have calculated Moment of inertia=square of bond lengths i.e something of the order of 10-20m2 multiplied by square of nuclear diameter or radius i .e something of the order of 10-30m2. Multiplying those values time nuclear masses .
so you mean he distance from the axis of rotation to the increment of mass=square of bond lengths multiplied by square of nuclear diameter or radius .
right?

 

[Bystander]

No. No. No.
I will give you a toothpick. You will stick one baby green pea on one end and a second baby green pea on the other end. The bond length is the length of the toothpick, or the distance between the peas. The peas are the two atomic nuclei in the diatomic molecule. Lay it on the table in front of you. You can spin it on the table (horizontal plane) around the center of the toothpick (the point midway between the peas), or you can roll it as if the toothpick were an axle and the peas were little bitty tires, or you can rotate it in a vertical plane about the center of the toothpick. These are the three axes of rotation. The moments of inertia for rotation in either the horizontal or vertical planes are calculated by squaring the distances of the peas from the center of the toothpick, multiplying the squares by the masses of the peas and adding them. The moment of inertia for rolling the toothpick is calculated by finding how far the peas are from the axis of the toothpick, and that is at most only half the diameter of the peas (the toothpick is stuck through them), squaring that distance and multiplying by the masses of the peas.

You do NOT square one distance for one moment and multiply it by the square of another distance for a different moment.

 

[gracy]

the axis of the toothpick,

what you mean by the axis of the toothpick?You have been very patient till now clearing all my silly doubts,thanks for it and i am just about to get it.

 

[Bystander]

Is a toothpick approximately a line segment? Think of the long dimension.

 

[gracy]

Is a toothpick approximately a line segment? Think of the long dimension.

Now,please tell me what you mean by axis of toothpick?

 

[Bystander]

Draw a line from the point at one end through the toothpick to the point at the other; strictly speaking that is the "major" axis.