Understanding Degrees of Freedom in Calculating Molecule Kinetic Energy

[elabed haidar]

i need to know how many degrees do i need to be able to find the kinetic energy of the molecule ?? and may i have a brief defintion of the degree of defintion i really need to know if I really understood the meaning of the degree of definition in class and thank you

 

[jambaugh]

It of course depends on the molecule.

The number of degrees of freedom is the number of unconstrained parameters you need to exactly specify the configuration of a molecule. For a diatomic molecule treated as a rigid bar-bell you would have 6 parameters (3 for each atom's coordinates) minus the 1 constraint (that they are a specific distance apart). So a total of 5 degrees of freedom.

You can count another way: three degrees of freedom for the position of the center of mass of the rigid molecule plus two degrees of freedom for the angular orientation (think azimuthal and inclination angles) for a total of 5. This in general is the degrees of freedom for any rigid body.


But typically at higher temperatures the inter-atomic bonds act as springs instead of as rigid rods and you get the full six degrees of freedom for a diatomic molecule.

For say water, you have potentially 9 degrees of freedom if the temperature is high enough to significantly induce vibrations in the hydrogen-oxygen bonds. But at typical temperatures of water vapor you would only have the 5 of a rigid rotator or possibly one more where the angle between the hydrogen atoms is allowed to vary.

The "at what temperature" issue is a matter of the quantized energies of these bond vibrations. Figure the energy of the first excitation and if that is significantly more than kT then you'll have no appreciable excitation and you can treat that mode of vibration as "locked out".

For some molecules the determination of degrees of freedom can get quite complex. For example Ethane as two carbons bonded with a single bond and each to three hydrogens. At normal temperatures the hydrogen bonds may be treated as rigid but the two H3C halves can pivot if the thermal energy is above the http://en.wikipedia.org/wiki/Ethane" .

 

[NEILS BOHR]

you need to specify the molecule first!

 

[elabed haidar]

thank you but the definition of the degree is still not clear unconstrained parameters ? what do you mean by that?
how can you the number of degrees needed in each molecule ? i know its different in each type and i know i need to know the molecule but how do i specify it? that is what i am still not understanding it

 

[jambaugh]

thank you but the definition of the degree is still not clear unconstrained parameters ? what do you mean by that?
how can you the number of degrees needed in each molecule ? i know its different in each type and i know i need to know the molecule but how do i specify it? that is what i am still not understanding it

I don't know what else to tell you besides give examples.

With molecules you can generally start with the 3 coordinates of each of the atoms and then carefully! count your constraints.

Start with the number of atoms, say 3. Then prior to applying constraints you have 3 spatial coordinates for each of the 3 atoms and that's 9 parameters. Now it's a molecule and not just 3 separate atoms going about their business... so there are constraints... how many depend on the bond. If say the molecules are bonded in a linear configuration that's two bonds between them so two constraints, assuming the bond angle is free to move (imagine a pair of nunchuks flipping around in space.) The constraints are how far apart each bonded pair must be. 9-2 = 7 free parameters.

Consider how these parameters specify the configuration. Use 3 coordinates (x,y,z) to specify the position of the middle atom in space. Now specify the position of each of the other two atoms using spherical coordinates with the middle atom as the origin. That's (r1,theta1, phi1) and (r2,theta2, phi2). But you know the bond length so r1 and r2 are not free parameters but constants. You can specify then exactly the configuration of the atom by giving the seven parameters: (x, y, z, theta1, phi1, theta2, phi2).

If instead you have the three atoms forming a triangular bond, each atom bonding to the other two, (or if the type of bond constrains the bonding angle of the middle atom) then you end up with one more constraint and one less degree of freedom, we're back to the 5 free parameters of a rigid rotator. These may be for example, the position of one atom (x,y,z), the angular position of the second relative to the first, (phi1,theta1) and then
at what angle (about the axis through the first and origin atoms). That's six.

NOTE: I erred in my earlier post when specifying five degrees of freedom for the rigid rotator. It has six. The diatomic molecule is NOT a rigid rotator as it's symmetry about the axis through the two atoms means rotation about that axis is not a degree of freedom for the molecule.

For the rigid rotator you can always start with a standard orientation centered at the origin. Then specify the translation to its actual position (x,y,z) and the vector rotation angle about its center (theta1, theta2, theta3). These six degrees of freedom are the parameters of the kinematic lie group for the object in question. The generators of this group are the canonical momenta which appear in the Hamiltonian of the system.

For single atoms you have only the 3 position degrees of freedom and for diatomic molecules you have the 5 I mentioned. For 3 or more you should start with the 6 degrees of freedom for a rigid rotator and then add any extras which come from deviations from rigidity, i.e. can atom A pivot freely? Then add one or two degrees of freedom depending on whether part of the "pivot freely" is already covered by rotating the whole molecule.

Well that's about all I can tell you. Each molecule type must be analyzed and double checked to see that you didn't over count and didn't miss any degrees of freedom. You also, as I've mentioned, may need to alter your analysis as energies increase sufficiently to "bend" your constraints. This is an exercise in quantum mechanics, where the elasticity of a constraint will tell you the amount of energy needed to excite the corresponding vibrational mode. When the temperature approaches that excitation energy (kT ~ E) you have to start adding that degree of freedom. See e.g. http://www.pci.tu-bs.de/aggericke/PC4e/Kap_V/H2O_Schwingungen.html" .

 

[jambaugh]

I got curious and found a table of specific heats and gas constants for various gasses online. I edited a spreadsheet and uploaded it to Google Docs:

https://spreadsheets.google.com/ccc?key=0Ann6rf9lnHMcdDh3SlJSN1k1a2ctQ0xuNHNhVC0yV0E&hl=en"

Notice how ideal gasses come out with 3 degrees of freedom, and most diatomic have around 5. Note also how water varies between 6 and 7 according to pressure. That's the vibrational degree of freedom manifesting.

Note also the exceptional behaviors of, Carbon disulphide, and Nitrogen tetroxide. They seem to have more degrees of freedom per atom than 3 times the number of atoms per molecule. This could be an error in the R value or there may be something peculiar going on such as higher Van der Wal's forces suppressing their R constants.

Fractional values for the DoF values indicate possible semi-rigid constraints/degrees of freedom as well as measurement imprecision.

I'm also curious about the 4.9 value for diatomic hydrogen. Possibly there's some mono-atomic hydrogen present or maybe exchange symmetry at the quantum level is having an effect. Or maybe it's just imprecision. I don't know the error bars on the numbers.

Regards,

 

[elabed haidar]

thank you very much but i have one thing what is the meaning of constraints ?? how do you specify them ? here is the main problem i know i seem dumb but i am really having problems in understanding it but thank you veyr much again

 

[jambaugh]

thank you very much but i have one thing what is the meaning of constraints ?? how do you specify them ? here is the main problem i know i seem dumb but i am really having problems in understanding it but thank you veyr much again

Think in terms of mechanical devices. Imagine a wheel (in the plane). It can turn and it can slide around. You pin down the axle and you are constraining the system. It can then only rotate. Mathematically you have parameters for the wheel x and y for the position of its center and theta for the angle it is rotated (imagine a reference mark on the wheel). The constraints above are say: x=3 and y=42. You've constrained the wheel by fixing its center at (3,42).

Another example. You imagine a particle moving in 3 dimensions. It has 3 parameters, (x,y,z) for its position. Now you decide to constrain it (say by connecting it via a 3" rod to a point you'll define as the origin). The constraint is r = sqrt(x^2+y^2 + z^2) = 3". You have three variables and one constraint in the form of an equation.

The constraint equation means only two of the three variables is a free parameter i.e. a degree of freedom. You can solve for say z in terms of x and y via z = sqrt(3^2 - x^2 - y^2). Thus the particle constrained to a fixed distance (say 3") from the origin has only two degrees of freedom. These can be its x and y coordinates (with z as a function of x and y) or you can use lattitude and longitude since that point can only move along a sphere of radius 3". Note that the range of motion defines a 2 dimensional surface (a sphere) which equates to the two degrees of freedom.

The degrees of freedom of a system will be fixed while you may use too many parameters to specify the system because you use free parameters for each component without imposing the constraints. How you do this can vary but you'll always have:
fixed: degrees of freedom = variable number of parameters - variable number of constraints.

Physically or geometrically you see a constraint as some type of connection between components. Algebraically you see it as equations imposed on the parameters used to describe each component.

In the particular case of molecules you generally see the inter-atomic bonds as strong constraints while there will be weaker constraints due to either quantum mechanically derived fixed bonding angles (as with water) or due to weaker interaction (typically repulsion but sometimes polar attraction) between atoms which are not directly bound to each other (as with say methane where the four hydrogens repell one another and try to stay in a tetrahedral configuration).

By "strong" or "weak" here I do not mean in the mathematical sense. (There are formal definitions for weak vs strong constraints mathematically). I am rather thinking in terms of relative strength of the forces of constraint.

When it comes to actual physical systems the constraints really take the form of a potential function of the parameters. It's not so much that the parameter has a fixed value but that it takes energy to change it from its equilibrium value. Example:

A particle in a central harmonic potential (force is attractive and proportional to distance from the center). The particle is not absolutely constrained, it can move arbitrarily far from the center given enough energy to pull it away. It has then 3 degrees of freedom (x,y, and z displacements).

But calculate the frequency at which the particle will vibrate, then apply quantum mechanics to determine the discrete energy levels (dE = hbar omega ) and you can treat the system as rigidly constrained (zero degrees of freedom) if the available energy is substantially below this quantum amount, i.e. if the temperature is significantly less than dE/K where K is Boltzman's constant.

 

[elabed haidar]

thank you very much i hope i wasnt a burden to you