# How are microstates formed in chemistry?

##### 1 Answer

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.

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 ** 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

Each one would be a linear combination

#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

#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:

- 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. #bb"Any number"# of particles (presumed indistinguishable)**could occupy the same compartment at the same time**(*bosons*, such as photons).