Proof: Partition Function of 3 Systems A, B, & C

[NewtonApple]

Homework Statement

For three systems A, B, and C it is approximately true that $Z_{ABC}=Z_{A}Z_{B}Z_{C}$. Prove this and specify under what conditions this is expected to hold.

Homework Equations

Z is the partition function given by $Z=∑e^{-ε/KT}$
ε is energy, T is temperature and K is Boltzmann constant.

The Attempt at a Solution

let say that A is the translational, B is the vibrational and C is the rotational energy levels for diatomic molecule.

To a good approximation the different forms of molecular energy are independent, so that we can write

$ε_{total}= ε_{A}+ε_{B}+ε_{C}$​

Since $Z=e^{-ε/KT}$, the sum in the exponents becomes a product.

$Z_{total}=(∑e^{-ε/KT})_{A}(∑e^{-ε/KT})_{B}(∑e^{-ε/KT})_{C}$

$Z_{ABC}=Z_{A}Z_{B}Z_{C}$​

But what will be the conditions?

 

[lightgrav]

You already mentioned one qualifier:

To a good approximation the different forms of molecular energy are independent, so that we can write

$ε_{total}= ε_{A}+ε_{B}+ε_{C}$​

When are the Energies in each different form NOT independent? (Think extremes, here!)

 

[haruspex]

You are right that independence is important, but I don't think you've used that assumption in the right way.
$ε_{total}= ε_{A}+ε_{B}+ε_{C}$ is true anyway. Don't you need the independence to get from
$Z_{total}=\Sigma_S e^{-ε_{tot}/KT}$
to
$Z_{total}=\Sigma_A \Sigma_B \Sigma_C e^{-ε_{tot}/KT}$
?
I.e. the microstates of the combined system are merely the combinations of the microstates of the separate systems.

 

[TSny]

I might not be interpreting the word "independence" the same as way others are. I think of it as meaning non-interacting.

Non-interaction of A, B, and C is important in being able to write ε_total = ε_A + ε_B + ε_C. An example where the energy cannot be written this way is a system with a potential energy of interaction U(A,B) between subsystems A and B.

Also, there are systems for which the subsystems are strongly interacting but yet the sum over microstates of the total system can still be written as a multiple sum over the microstates of the subsystems. For example, consider a system of 3 interacting spins (A, B, and C) as in the 1D Ising model (See here, especially slide 5). (But the partition function of the total system does not factor into a product of individual partition functions due to the fact that the energy cannot be written as ε_total = ε_A + ε_B + ε_C .)

Even for a system of three non-interacting particles A, B, and C (e.g., three non-interacting particles in a box), there is an important situation where the partition function does not factor as Z = Z_A Z_B Z_C. Think about the case where the particles are indistinguishable. Note that Z doesn't factor even though ε_total = ε_A + ε_B + ε_C.

 

[HalfLostAndSearching]

This approximation will hold true when the systems A, B, and C are not interacting with each other and their energy levels are independent. This is usually the case for simple systems such as diatomic molecules, where the translational, vibrational, and rotational energy levels can be treated as independent. However, for more complex systems, such as molecules with multiple atoms or systems with strong interactions, this approximation may not hold and the partition function will need to be calculated using more advanced methods.