Maksvell-Bolsman, Boze-Eynshteyn va Fermi-Dirak funksiyalarining taqsimlanishi

Maksvell-Bolsman, Boze-Eynshteyn va Fermi-Dirak funksiyalarining taqsimlanishi
Termodinamikaning 1- va 2-qonunlarining birlashgan tenglamalarini qo’llash dU va dS kattaliklarning to’liq differensial ekanligiga asoslangan. Bu moddalarning xossalarini tavsiflovchi turli kattaliklar orasida o’zaro bog’liqlikni topishga imkoniyat beradi. Termik va kalorik koeffisiyentlar o’rtasidagi bog’liqlikni ko’rib chiqamiz.
Mustaqil o’zgaruvchilar deb V va T larni qabul qilamiz,
unda Q = dV+C_v dT (I.40)
bu yerda: va S_v–kalorik koeffisiyentlar; = (∂Q/∂V)_t –izotermik kengayish issiqligi; S_v = (∂Q/∂T)_v –o’zgarmas hajmdagi issiqlik sig’imi. Ushbu qiymatni termodinamikaning 1- va 2-qonunlarining tenglamalariga qo’ysak,
dU = δQ - pdV = C_vdT + ( -p)dV (I.41)
va dS = δQ/T = dV/T+C_vdT/T (I.42)
dU va dS larning to’liq differensialligini hisobga olib, quyidagicha yozish mumkin (xususiy hosilalarning yig’indisi to’liq differensialni beradi, ularning o’rnini almashtirish mumkin). (I.41) dan:
(∂S_v /∂V)_t = ∂( p)_v/∂T = ( / T)_v( p/ T)_v (I.43)
(I.42) dan: ( / T)( /T)_v= (C_v/T)_t/ V yoki [T( / T)_v ]/T=( C_v/ V)_t ( I.44)
bundan: ( C_v/ V)_t=( /T)_v /T (I.45)
(I.43) dan (I.45) ga ( C_v/ V)_t ning qiymatini qo’yamiz:
( / T)_v( p/ T)_v=( / T)_v /T (I.46)
bundan: =T( p/ T)_v (I.47)
yoki  =( /p₀)( p/ T)_v; = pT.
Ideal gaz uchun ( p/ T)_v=R/V bo’lgani bois (I.47) dan:
=RT/V = p (I.48)
(I.48) va (I.40) tenglama quyidagi ko’rinishni oladi:
δQ = pdV+C_v dT (I.49)
(I.46) tenglamani differensiallaymiz:
( / T)_v = ( p/ T)_v+T( ²p/ T²)_v (I.50)
(I.50) va (I.45) tenglama bilan solishtiramiz:
( C_v/ V)_t = T( ²p/ T²)_v (I. 51)
(I.51) tenglama C_V ning hajmdan bog’liqligini beradi.
Uni ideal gaz uchun qo’llab, p = RT/V; ( p/ T)_v = R/V va ( ²p/ T ²)_v = 0, bundan ( C_v/ V)_t = 0 ekanligi, ya’ni ideal gazning issiqlik sig’imi hajmdan bog’liq emasligi kelib chiqadi.
Mustaqil o’zgaruvchilar r va T bo’lganda,
Q = hdp+C_p dT (I.52)
va dS = hdp/T+C_pdT/T (I.53)
bundan ∂(h/T)_p /∂T = /T (∂C_p /∂p)_t (I.54)
yoki (∂h/∂T)_ph/T = (∂C_p /∂p)_t (I.55)
hosil bo’ladi.
Birinchi qonunga binoan h= (∂V/∂p)_t ekanligini e’tiborga olib, (I.46) va (I.47) tenglamalarni hisobga olsak,
h = T( p/ T)_V(∂V/∂p)_t (I.56)
Ammo, ( p/T)_V(∂V/∂p)_T(∂T/∂V)_p =  , undan ( p/ T)_V(∂V/∂p)_t = (∂V/∂T)_p va h = T(∂V/∂T)_r = -VT (I.57)
Ideal gaz uchun (∂V/∂T)_r = ∂/∂T(RT/p)_p=R/p, shuning uchun h=RT/p=-V va Q=C_p dT-Vdp. (I.57) tenglamani T bo’yicha differensiallaymiz:
(∂h/∂T)_p=-T( ²V/ T²)_p( V/ T)_p.
U holda (I. 55) tenglamadan foydalanib,
( S_p / p)_t = -T( ²V/ T ²)_p (I.58)
tenglamani olamiz.
Ideal gaz uchun ( ²V/ T²)_p = 0 va ( S_p / p)_t = 0, ya’ni ideal gazning S_r si bosimga bog’liq emas.
Xuddi shunday mustaqil o’zgaruvchilar sifatida sistemaning boshqa 2 ta parametrini olib (masalan, V va r), barcha kalorik va termik koeffisiyentlar orasida o’zaro bog’lanishni topish mumkin.


I.8. Ideal gazning turli jarayonlardagi kengayish ishi, jarayon issiqligi va ichki energiyaning o’zgarishi

Izobarik jarayon uchun:

W_p = pV = nR(T₂-T₁) (I.59)
Q_p = nC_p(T₂-T₁) = H (I.60)
U = nC_v(T₂-T₁) (I.61)
O’zgarmas bosimda sistemaga berilgan issiqlik sistemaning ichki energiyasini o’zgartiradi va ish bajaradi:
Q_p = dU+pdV (I.62)
dU = C_V dT (I.63)
Q_p = C_v dT+pdV (I.64)
Izotermik jarayon uchun:
W_t = nRTln(V₂/V₁) = nRTln(p₁/p₂) (I.65)
Q_t = W_t; (I.66)
U = 0 (I.67)
Adiabatik jarayon uchun:
W_S = U (I.68)
U = nC_V(T₂-T₁) (I.69)
Q_s = 0 (I.70)
W_s = nC_V(T₁-T₂) (I.71)
Izoxorik jarayonda kengayish ishi bajarilmaydi
W_V = 0 (I.72)
ideal gazning ichki energiyasi faqat haroratning funksiyasidir. Joul qonuni bo’yicha (∂U/∂V)_t = (∂U/∂p)_t = 0; U_t = const (I.73)
Ichki energiyaning o’zgarishi izobarik va izoxorik jarayonlarda bir xil bo’ladi: U = nC_v(T₂-T₁) (I.74)
Jarayon issiqligi ichki energiyaning o’zgarishiga teng bo’ladi:
Q_v = U = nC_v(T₂-T₁) (I.75)


I.9. Puasson tenglamalari
Ideal gazning adiabata tenglamasini chiqarish uchun Q = rdV+C_VdT dan Q = 0 bo’lganligi sababli, -nC_VdT = pdV tenglamadan:
rdV+C_VdT = 0 (I.76)
(I.76) ga p = RT/V qo’yib, T ga bo’lsak, (RdV/V)+C_VdT/T = 0 va R = C_p-C_V bo’lgani uchun (C_p-C_V)dV/V+C_VdT/T = 0 (I.77)
(I.77) ni C_V ga bo’lib, C_p/C_V = γ deb belgilaymiz:
(γ-1)dV/V+dT/T = 0 (I.78)
(I.78) ni integrallasak,
lnV ^{γ -1}+lnT = const yoki TV ^{γ -1} = const (I.79)
hosil bo’ladi.
Xuddi shu yo’l bilan Tp^{(1- γ)/ γ} = const (I.80)
tenglamasini chiqaramiz.(I.79) ni (I.80) ga bo’lsak, pV ^γ = const (I.81)
ni olamiz. (I.79), (I.80) va (I.81) tenglamalar Puasson tenglamalari deyiladi.