The probability of finding the system in microscopic state [itex]i[/itex] is:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]p_{i}=\dfrac{1}{Q}e^{-\beta E_{i}}[/itex]

Where [itex]Q[/itex] is the partition function.

Assumption: molecule [itex]n[/itex] occupies the [itex]i_{n}[/itex]th molecular state (every molecule is a system).

The total energy becomes:

[itex]E_{i_{1},i_{2},...,i_{N}}=\epsilon_{i_{1}}+ \epsilon_{i_{2}}+...+\epsilon_{i_{N}}[/itex]

[itex]Q=\underset{i_{1},i_{2},...,i_{N}}{\sum}e^{-\beta\left(\epsilon_{i_{1}}+ \epsilon_{i_{2}}+...+\epsilon_{i_{N}}\right)}[/itex]

[itex]=\underset{q}{\underbrace{\left(\underset{i_{1}} {\sum} e^{-\beta e_{i_{1}}}\right)}}\times\left(\underset{i_{2}} {\sum} e^{-\beta\epsilon_{i_{2}}}\right)\times...\times\left(\underset{i_{N}}{\sum}e^{-\beta\epsilon_{i_{N}}}\right)[/itex]

Where [itex]q[/itex] is the molecular or particle partition function.

The partition function becomes [itex]Q=q^{N}[/itex] . This is valid for distinguishable particles only (why?).

The probability of finding molecule [itex]n[/itex] in molecular state [itex]i'_{n}[/itex] is obtained by summing over all system-states subject to the condition that [itex]n[/itex] is in [itex]i'_{n}[/itex]

[itex]p_{i'_{n}}=\dfrac{1}{Q}\underset{i_{1},i_{2},...,i_{N}}{\sum}\delta_{i_{n},i'_{n}}e^{-\beta\left(\epsilon_{i_{1}}+ \epsilon_{i_{2}}+...+ \epsilon_{i_{N}}\right)}=\dfrac{1}{Q}e^{-\beta\epsilon_{i'_{n}}}q^{N-1}=\dfrac{1}{q}e^{-\beta\epsilon_{i'_{n}}}[/itex]

So why is [itex]Q=q^{N}[/itex] only true when the particles are distinguishable and what does it mean when it is stated that "the probability of finding molecule [itex]n[/itex] in molecular state [itex]i'_{n}[/itex] is obtained by summing over all system-states subject to the condition that [itex]n[/itex] is in [itex]i'_{n}[/itex]"

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# Molecular partition function

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