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
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#(+1" for particle exchange")#
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#(+1" for particle exchange")#
#barul|" "color(white)(cdot)" "|barul|" "cdot" "|" "bb((5))#
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#(+1" for particle exchange")#
#barul|" "cdot" "|barul|" "color(white)(cdot)" "|" "bb((6))#
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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).
