# Rotational partition function for CO2 molecule

• DannyJ108
In summary, the conversation discusses the calculation of the rotational partition function for a CO2 molecule. There are two formulas mentioned, one for diatomic molecules and one for polyatomic molecules. It is mentioned that the first formula is only valid for diatomic molecules, but it is also used for linear polyatomic molecules like CO2. The second formula is for asymmetric polyatomic molecules and is not applicable to CO2. There is confusion about which formula to use and the experts are asked for help.

#### DannyJ108

Homework Statement
Find the rotational partition function for a CO2 molecule. Assume ideal gas and classical approximation
Relevant Equations
##\zeta^r= \frac T {\sigma \theta_r}##
Hello fellow physicists,

I need to calculate the rotational partition function for a CO2 molecule. I'm running into problems because I've found examples were they say this rotational partition function is:

##\zeta^r= \frac T {\sigma \theta_r} = \frac {2IkT} {\sigma \hbar^3}##

Where ##\zeta^r## : rotational partition function for the molecule
##T## : temperature
##\sigma## : symmetry number
##\theta_r## : characteristic rotational temperature
##I## : moment of inertia
##K## :Boltzmann factor

The problem is that I've read that ##\zeta^r## mentioned before is ONLY valid for DIATOMIC molecules. There's another more complicated formula used for polyatomic molecules:

##\zeta^r= \frac {\left( \pi I_1 I_2 I_3 \right)^{1/2}} {\sigma \hbar^3} \left( 2kT \right)^{3/2}##

But I've read this formula is for asymmetric polyatomic molecules with different inertia moments, whereas CO2 only has one moment of inertia.

Now I don't know what formula to use. Can you guys help me out?

This is a link (http://myweb.liu.edu/~nmatsuna/PHS702/statmech/lect7/stat.mech.8.html) for polyatomic molecules and it says that the ##\zeta^r## for a linear polyatomic molecule (which is the case of CO2) is the first formula I wrote out.

This other one (http://faculty.washington.edu/gdrobny/Lecture453_17_2013.pdf) says that ##\zeta^r## is only valid for diatomic molecules, but then proceeds to write the partition function with that formula.

Regards.

DannyJ108 said:
This other one (http://faculty.washington.edu/gdrobny/Lecture453_17_2013.pdf) says that ζr is only valid for diatomic molecules, but then proceeds to write the partition function with that formula.
It says that the formula it gives for the total partition function is only valid for diatomics; the difference for linear polyatomics is in the vibrational component, not the rotational.

## 1. What is the rotational partition function for a CO2 molecule?

The rotational partition function for a CO2 molecule is a mathematical expression used to describe the distribution of energy among the different rotational states of the molecule at a given temperature.

## 2. How is the rotational partition function calculated?

The rotational partition function is calculated by summing over all possible rotational states of the molecule, weighted by their respective energies and Boltzmann factors.

## 3. Why is the rotational partition function important?

The rotational partition function is important because it allows us to calculate thermodynamic properties of a molecule, such as its heat capacity and entropy, which are crucial for understanding its behavior in various conditions.

## 4. How does the rotational partition function change with temperature?

The rotational partition function increases with temperature, as higher temperatures allow for more rotational states to be populated and contribute to the overall energy distribution of the molecule.

## 5. Can the rotational partition function be applied to other molecules?

Yes, the rotational partition function can be applied to any molecule that has rotational degrees of freedom, such as diatomic molecules or linear molecules like CO2. It is a fundamental concept in statistical thermodynamics and is used to study the behavior of various molecules in different environments.