Question

2) Next week, we will show that the partition function for a monatomic ideal gas is given by Q(N,V,T) - 1 ( 2mk,T 30/2 ? N 422) VN where m is the mass of the gas molecules and h is Planck's constant. Derive expressions for the pressure and energy from this partition function.

Answer 1

. have 7 we have the perstition function for monoatomie ideal & (NVT) (2Tim KRT 23% . for simplicito Let is consider. (2 m ) v = q (VF). .. g (N, V,F) = (a cv, f)] This is done for over coming the mathematical difficulty of upcoming problems. . Ing = sing – lnN! = - 3 haß +31 lm (25 ) Nohava hill – – ₂. luß & other terms involving NAV 2-3 Com - - NK&T NKBT (.....) Derivation of the expression of pressure Pressure of a macroscopic system is given by P. (NV) = - ( 2 ) e average Pressure (P) is given by (P>= [ki (Ny v, f) Pj (N, 4) P (NV) - Pressure if ith state B. (N. V, B) = Probability of finding the Perticale inth Ithstate

.. (P) = 2 by Cwmps (-20). (-) BE ..[ b, Now, Let us evaluate the vaheot (29) N.B. av UNIB - {{e pes} --ß I (27) é Bej GB e equation ③ be comes - <P>= Blog DNB - 1 (o lang niß Now we have to evaluate the value at (or lua) know, ale = tlna – kan! – – 2 leß + 30 lm (275) + NhV – bunl we know, <P>= 12 V .: <P> - NKBT <P = R ORT [na?

Derivation of average energy (LE)) The average energy of a system is given by <E) = {P. E j = energy of the ith state P = Probability of finding Pesticle in the state. .: (E) = E é .... i 0 Now, set us evaluate the value of (ao) ( Der er ste pes} N , =-{ €; e BE equation (in, can be rewritten an- <F) --25. In a a B INV OB VN from equation is we have, (aha) = -3 NKET. LE) = 32 N K&T - ERT For no mole moncatonnie idealgan. (E)= ORT