Question

5. The number of complexions (2) of preparing a system and its entropy (S) are related through the Boltzmann constant (k) as a. S = k in 2 b. S = k In 1/92 C. S = In 2 d. S = -k In 12 e. none of these m! 8. The entropy of N particles distributed into m boxes is a. S = min m! N!(m-N)! b. S = p In- N!(m-N)! C. S = n In N!(m-N)! d. S = k In m! N!(m-N)! e. none of these m! 9. The change in entropy (AS) when N particles occupying mi identical boxes increases to m2 identical boxes is a. AS = k N (In m2 - Inm) b. AS = k N In(mz/m) C. AS = k N (In m2 + Inm) d. AS = k N (In mı - In m2) e. none of these
please explain why chose those answer for the 3 questions.

Answer 1

solution : the In no. Stortistic micro-states the be if and entropy (S) micro-states is . Boltzmann- of. relation between and number. bu the 1 given S = In 2 K.. In constant. thé Boltzmann is Here, K rs option @. the answer So, for this ways distributing boxes ff ticles is; m 8 Now, homber pan in N m! in CN WE N! (MN)! micro-states so here is, number of m! =W NI (m-N)! particle of N boxes entropy into ic soi The distributed m m! Koln N! (m_M)] S= Kilne

a mi boxes, 9. panticle Occupying Now, N entropu, Si=kin mi! N! ( MM) boxes m2 occupying and if. N particle entropy S=kih M2! N! ( MN)! entropy So, Change in si AS Sz mi! -kin M₂! Kiln NI (M ₂ ~ NJ! N! (M-M)] [In M, ! – Inx! - in cm - 1)! - In th, ! + Inn! + In (9,- 1.! w!? kl Inma! - in (M₂ - N)! inmi! tin (m, -1)! z mation, approxi! Now, by stirling in m! m mlnm ma So, in ma! M - In my! (h, — N) mai in me m. ln m In (1-1)! =(M)- N) (M= N) In(M-N)! = cm, -N) 94( MM) cm, -a)

valves After putting this AS = K = ['m, 1 May - - Me -(13-N) In (m, - ) + (M2-il) Mr. In my + m + 6M-N) In[^,- ) (m, n) = K[m. In my CM-N) in (m_M) minm, t cm, -N) in 6m, -N) - k[m2In my m; In CM2 - x) + N. 1n (m2-n) meln mi + my.lacme-w) at Niin (m, -1) ,-] we Gan NOW, for mi, ma a N Ma- Writer no 2 ni MN so, As =*[m3 In my -- mIn my + N. In the — M-10m, + minim, - Nilnm = KN. [Inma KN in. (m. In ( ) = 1 nm - ?nmi om in mi as, [ m, So, when N 2 particles Occupying mi to increases boxes boxes in entropy, Change js = KNIH (ME) = KN. [in my - Inmi