### How do you find the degrees of freedom

Posted:

**Sun Jan 24, 2016 6:58 pm**What exactly are the degrees of freedom for a gas molecule and how would one calculate this number for diatomic and monatomic ideal gases?

Created by Dr. Laurence Lavelle

https://lavelle.chem.ucla.edu/forum/

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=128&t=10567

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Posted: **Sun Jan 24, 2016 6:58 pm**

What exactly are the degrees of freedom for a gas molecule and how would one calculate this number for diatomic and monatomic ideal gases?

Posted: **Sun Jan 24, 2016 8:11 pm**

The degrees of freedom are the way something can move in space (pg 276).

Something that is amorphous will have 6 DoF's, because it can translate in three directions, and each rotation about one axis will have it "look" different than when it started.

So a perfect sphere will only have the three translations because if it's rotated about any of the axes it will look exactly the same.

For a linear molecule, 1 axis of rotation will have it look exactly the same, so 6-1=5 DoF's.

For a bent molecule, a rotation (anything between 0---->360 in each of these cases) about any axis will have it look different. 6-0= 6 DoF's.

Multiply that number by 1/2k*T an you have the kinetic energy for the atom (KE).

U=nR*KE

It's probably easier to memorize

monoatomic gas U=3/2nRT

Diatomic linear gas U=5/2nRT

(Which has a relationship with C_{v} as well).

Something that is amorphous will have 6 DoF's, because it can translate in three directions, and each rotation about one axis will have it "look" different than when it started.

So a perfect sphere will only have the three translations because if it's rotated about any of the axes it will look exactly the same.

For a linear molecule, 1 axis of rotation will have it look exactly the same, so 6-1=5 DoF's.

For a bent molecule, a rotation (anything between 0---->360 in each of these cases) about any axis will have it look different. 6-0= 6 DoF's.

Multiply that number by 1/2k*T an you have the kinetic energy for the atom (KE).

U=nR*KE

It's probably easier to memorize

monoatomic gas U=3/2nRT

Diatomic linear gas U=5/2nRT

(Which has a relationship with C