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.