How Does Electron Spin Affect the Partition Function in Saha's Equation?

In summary: This equation states that the number of states (represented by letter s) equal the number of possible values of the partition function (Z_int), which is equal to the number of states the electron can have when it is bounded by the proton (Z = 2).
  • #1
Sebas4
13
2
Hey, I have a question about proving Saha's equation for ionizing hydrogen atoms.
The formula is
[tex] \frac{P_{p}}{P_{H}} = \frac{k_{B} T}{P_{e}} \left(\frac{2\pi m_{e} k_{B}T}{h^2} \right)^{\frac{3}{2}}e^{\frac{-I}{k_{B} T}} [/tex]
with
[itex] P_{p} [/itex] pressure proton's,
[itex] P_{H} [/itex] pressure hydrogen atoms,
[itex] m_{e} [/itex] mass electron,
[itex] T [/itex] temperature resevoir,
[itex] I [/itex] ionization energy,
[itex] k_{B} [/itex] Boltzmann's constant,
[itex] h [/itex] Planck's constant.

When the electron is bounded, the electron has two spin states.
So the three situations are, the electron is bounded and is in one of the spin states, the electron is bounded and is in the other spin state, and the electron is not bounded by the proton.
The system of interest is the proton.
When the electron is bounded by the proton, the energy of the system of interest is [itex]-I[/itex].
I have derived that
[tex] \frac{P_{p}}{P_{H}} = \frac{1}{2}\frac{k_{B} T}{P_{e}} Z_{int} \left(\frac{2\pi m_{e} k_{B}T}{h^2} \right)^{\frac{3}{2}}e^{\frac{-I}{k_{B} T}} [/tex]
with [itex] Z_{int} [/itex] the partition function (for rotational/vibrational motion) of the electron is.
If you compare the two equations, you see that [itex] Z_{int} = 2 [/itex].
I don't get why [itex]Z_{int} [/itex] is 2.

The partition function is
[tex] Z = \sum_{s}^{} e^{\frac{-E\left(s \right)}{k_{B} T}} [/tex]
summing over all states (represented by letter s).
The partition function is not all the possible states the electron can have right?

So how does [itex] Z = \sum_{s}^{} e^{\frac{-E\left(s \right)}{k_{B} T}}[/itex] is equal to the number 2 (assuming that two spin states the electron can have, have non zero energy)?
 
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  • #2
In the absence of external fields, there is no energy associated with the spin state, so ##E(s) = 0## for all spin states. Therefore,
$$
Z_\mathrm{int} = \sum_s e^{-E(s) / k_B T} = \sum_s 1 = 2
$$
 
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FAQ: How Does Electron Spin Affect the Partition Function in Saha's Equation?

What is the partition function rotation?

The partition function rotation is a concept used in statistical mechanics to describe the distribution of energy states in a rotating system. It takes into account the rotational energy of molecules in a gas or liquid, and is an important factor in understanding thermodynamic properties of these systems.

How is the partition function rotation calculated?

The partition function rotation is calculated by summing over all possible energy states of a rotating molecule, taking into account the degeneracy of each state. This can be done using mathematical equations or by using computer simulations.

What is the significance of the partition function rotation?

The partition function rotation is important because it allows us to calculate thermodynamic properties such as heat capacity, entropy, and free energy for rotating systems. It also helps us understand the behavior of gases and liquids at different temperatures and pressures.

How does the partition function rotation differ from other partition functions?

The partition function rotation is specific to rotating systems and takes into account the rotational energy levels of molecules. Other partition functions, such as the translational and vibrational partition functions, describe the distribution of energy in other types of systems.

What factors can affect the partition function rotation?

The partition function rotation can be affected by various factors such as temperature, pressure, and the shape and size of the molecule. It can also be influenced by the presence of other molecules or external fields, such as electric or magnetic fields.

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