Question

1. Consider a system composed of a collection of N molecules that can distribute so that there are ni molecules in the state with energy E, n2 molecules in the state with energy E2, n3 molecules in the state with energy E..etc. The number of ways, W, in which molecules can distribute this way is N! N! W П "! n,in In,.. II Suppose that this system is undergoing the following process (at constant pressure and temperature): a. Final State: Initial State: The N molecules are distributed The N molecules are between all states, so that there are all in the E ni in the state E That is to say, n1=N and n2-n3n4= na in the state E2 ns.-0 na in the state E3 n4 in the state E4 stc. From the Boltzmann equation, Sk InW, derive the expression for the change in entropy for the process AS kN, In Xx Where Xi is the fraction of molecules in the state Ei (That is to say Xni N). To derive this equation you will need the Stirling approximation. In NI- N In N- N

This is a Physical Chemistry problem. Thank you in advance!

Answer 1

W- n! !... NIO!!!... Si = k ln Wi. Wq. = NI Stak ln Wf =ken No ...) =k[ln N! - (lasi! + Inne! t...)) Nylon very touge we can ase sterling approximation In a = alna-n Soak (NON-N-Baho: (0, 600,-n, + H2 b 22-D₂ +2 3 2 2 3 - Dat... ) k(wenn-fo - (n. enn, thanat.... - (n. fret not...))) Ek [Nen N &Bienni] = Volunt in het langt...) 5 {x; luni) KN ( A - Exikani] km Enilang_Nenn] *****Emin manat = kh({ni lnni - NN) = -k [n, enn,+ na In Na+ na lang - 19,+na+os+..) lemn] = ok mienn, -n, en N + n2nn-Dz en Nt....] -k in, en + n ln 21 +...] = -KN (EN enn ] = -KN EX; ln xi AS = St-Si = -KN Ex; lax; -o = -KN EX; lnxi