Question

How are microstates formed in chemistry?

Answer 1

Automatically?

Macrostates can be observed, even with quantum-sized particles, and one can extract microstates consistent with those macrostates simply by specifying the macrostate and using statistical predictions to enumerate the microstates.

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A microstate is one state (of many) for an ensemble of particles that is present in a distribution of possible configurations of a system, that all collectively describe a macrostate.

So, let's say a macrostate was described by the following statement:

Two particles placed in a four-compartment box, such that neither particle is in the same compartment.

Then, if the compartments are distinguishable, and the particles are distinguishable, there are `bb12` possible microstates:

`barul|" "cdot" "|barul|" "color(white)(cdot)" "|" "bb((1))`
`ul|" "cdot" "|ul|" "color(white)(cdot)" "|`
`(+1" for particle exchange")`

`barul|" "cdot" "|barul|" "cdot" "|" "bb((2))`
`ul|" "color(white)(cdot)" "|ul|" "color(white)(cdot)" "|`
`(+1" for particle exchange")`

`barul|" "color(white)(cdot)" "|barul|" "cdot" "|" "bb((3))`
`ul|" "color(white)(cdot)" "|ul|" "cdot" "|`
`(+1" for particle exchange")`

`barul|" "color(white)(cdot)" "|barul|" "color(white)(cdot)" "|" "bb((4))`
`ul|" "cdot" "|ul|" "cdot" "|`
`(+1" for particle exchange")`

`barul|" "color(white)(cdot)" "|barul|" "cdot" "|" "bb((5))`
`ul|" "cdot" "|ul|" "color(white)(cdot)" "|`
`(+1" for particle exchange")`

`barul|" "cdot" "|barul|" "color(white)(cdot)" "|" "bb((6))`
`ul|" "color(white)(cdot)" "|ul|" "cdot" "|`
`(+1" for particle exchange")`

If the particles are indistinguishable, then we have an extra (redundant) microstate which would be due to their exchange, and we only have `bb6` microstates possible here.

Each one would be a linear combination `psi_i` of the identical-looking configuration pairs `(phi_(ia),phi_(ib))`:

`psi_1 = c_1phi_(1a) + c_2phi_(1b)`
`psi_2 = c_3phi_(2a) + c_4phi_(2b)`
`psi_3 = c_5phi_(3a) + c_6phi_(3b)`
`psi_4 = c_7phi_(4a) + c_8phi_(4b)`
`psi_5 = c_9phi_(5a) + c_10phi_(5b)`
`psi_6 = c_11phi_(6a) + c_12phi_(6b)`

where the coefficients `c_i` describe the contribution the state makes to the distribution given by `psi_i`.

If the compartments are ALSO indistinguishable (but the boxes, stay connected in seemingly the same way), then...

• configurations `(1)`, `(2)`, `(3)`, and `(4)` (meaning `psi_1, psi_2, psi_3, psi_4`) are duplicates of each other (accounting for `90^@` and `180^@` rotational permutations, keeping the compartments sitting in the same orientation).
• configurations `(5)` and `(6)` (meaning `psi_5, psi_6`) are duplicates of each other (accounting for `90^@` rotational permutations, keeping the compartments sitting in the same orientation).

As a result, with indistinguishable compartments AND particles, we only have `bb2` nonredundant microstates `Psi_i` consistent with the given macrostate:

`Psi_A = c_13psi_1 + c_14psi_2 + c_15 psi_3 + c_16 psi_4`

(particles in adjacent compartments)

`Psi_B = c_17 psi_5 + c_18 psi_6`

(particles in diagonal compartments)

Examples of other complications might be:

1. The particles (presumed indistinguishable) that occupy the same compartment at the same time must be of opposite spin (fermions, such as electrons), restricting us to `bb"two"` particles maximum per compartment.
2. `bb"Any number"` of particles (presumed indistinguishable) could occupy the same compartment at the same time (bosons, such as photons).