変数分離dTc=T<latexit sha1_base64="bpcA4bypBLar//opb2adIFugBHs=">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</latexit>

dVV

(*)

左辺はTだけの関数で，右辺はVだけの関数である。
いかなるT,Vでも等号が成立するためには，左辺および右辺はある定数でAないといけない。

* 断熱過程ではdQ = 0なのでdU = dQ(= 0) + dW = dW = -PdVとなり，dU= CVdT なので，dU = CVdT = -PdV = -nRT / V dV, 両辺をnRT で割って，(CV / nR) dT / T = -dV / V, c dT / T = -dV / VVVf(Tf(T))

一般の関数f(T), g(V)の場合

g(V)
g(V )

TT

V
V

f(Tf(T))

一般の関数f(T), g(V)の場合

g(V)
g(V )

TT

V
V

一般の関数f(T), g(V)の場合

ここ1点のみで成立

f(Tf(T))

g(V)
g(V )

TT

f(T), g(V)がともに定数Aである場合

VV

g(V)
g(V )

f(T))
f(TT

f(T), g(V)がともに定数Aである場合

VV

=定数A
g(V)
g(V )

f(T))
f(TT

f(T), g(V)がともに定数Aである場合

VV
すべてのT,Vで成立

=定数A
g(V)
g(V )

f(T))
f(TT

dTdV
c==ATVd ln T
1dT
= , d(ln T ) =dTTTd(c ln T ) = d( ln V ) = A<latexit sha1_base64="EZKPoi8y2T/dpx1Vr/ONXym2GcI=">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</latexit>

(定数）

(定数）

d(clnT)= dc×lnT+ cd(lnT) = 0×lnT + cd(lnT)d(-lnV)= d(-1)×lnV - d(lnV) = 0×lnV - d(lnV)

y2

dy
dy
y1x1

dxdxx2<latexit sha1_base64="HwsnJ7Ua6NKH02HtoyG0uiE9Ze0=">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</latexit>

dx = d(c ln T ), dy = d( ln V )とおく

dx = dy = A (定数）となる。これを，以下のように図であらわす。

y2dydyy1x1

dxdxx2<latexit sha1_base64="HwsnJ7Ua6NKH02HtoyG0uiE9Ze0=">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</latexit>

dx = d(c ln T ), dy = d( ln V )とおく

dx = dy = A (定数）となる。これを，以下のように図であらわす。

y2

短冊状に足してゆくとdydyy1x1x2dxdxx2x1 = y2c ln T2y1c ln T1 = ln V2T2V1c ln= ln
T1V2⇥cT2V1ln= lnT1V2⇥cT2V1=T1V2cc
V1 T1 = V 2 T2( ln V1 )y2

短冊状に足してゆくとdydyy1x1x2dxdxx2x1 = y2c ln T2y1c ln T1 = ln V2T2V1c ln= ln
T1V2
⇥cT2
V1ln= ln
T1V2
⇥cT2
V1=T1V2cc
V1 T1 = V 2 T2( ln V1 )y2

短冊状に足してゆくとdydyy1x1x2dxdxx2x1 = y2c ln T2y1c ln T1 = ln V2T2V1c ln= ln
T1V2⇥cT2V1ln= ln
T1V2⇥cT2
V1=T1V2cc
V1 T1 = V 2 T2( ln V1 )y2

短冊状に足してゆくと
dydy
y1x1x2dxdxx2x1 = y2c ln T2y1c ln T1 = ln V2
T2
V1c ln= ln
T1
V2
⇥cT2
V1ln= ln
T1V2
⇥cT2
V1=T1V2cc
V1 T1 = V 2 T2( ln V1 )y2

短冊状に足してゆくとdydyy1x1x2dxdxx2x1 = y2c ln T2y1c ln T1 = ln V2T2V1c ln= ln
T1V2⇥cT2V1ln= ln
T1V2⇥cT2
V1=T1V2cc
V1 T1 = V 2 T2( ln V1 )y2

短冊状に足してゆくとdydyy1x1x2dxdxx2

x1 = y2c ln T2y1c ln T1 = ln V2
T2
V1c ln= ln
T1
V2
⇥c
T2
V1ln= ln
T1
V2
⇥c
T2
V1=T1
V2cc
V1 T1 = V 2 T2( ln V1 )

別解（こちらがより一般的）
T: T1 →T2
V: V1→V2<latexit sha1_base64="O/l5nZfbWyhPOHQHxVXXly493VQ=">AAAEeHicjZHLbtNAFIZP3UBLuDQtGyQ2I6KWRKLROKqgQkKqYMOyl8StFKeWM52kVh3bsieRUssvwAuwYIFAQlDxGGx4ARZ9BMSySEiIBWfGJpdGbXEU+8yZ/zv/P3YrcJ1IUHo6o83mrl2fm7+Rv3nr9p2FwuKSEfm9kPE6810/3GvZEXcdj9eFI1y+F4Tc7rZcvts6eiH3d/s8jBzfq4lBwJtdu+M5bYfZAlvWorbETMcTVlyz9GQf79WEmO3QZrGexLWExGbYJQdJjayQZ4SsplpDaY0JrTHUGsQ08xdNzTSm65Fa8m/13z5jtDGkzzsT0+Vt0SDKg5ih0zkUzYko0mN1XGaMZGOe6bRSOsiqKkRWOilnI0oKHm4ZaivL4GWZpaMyV8yoL42UzyNipm2ZpzRFpcnK+0zSk6S0upK6xFXmRbUaLU+BONZ5q1CkFaouMl3oWVGE7Nr0Cx/BhAPwgUEPusDBA4G1CzZE+GuADhQC7DUhxl6IlaP2OSSQR7aHKo4KG7tHeO/gqpF1PVzLmZGiGbq4+A+RJLBMv9ETeka/0s/0O/1z4axYzZBZBvhspSwPrIVX93Z+XUl18SngcERdmllAG9ZVVgezB6ojT8FSvn/8+mzn6fZyvELf0x+Y/x09pV/wBF7/J/uwxbffgPwA+vnXPV0Y1Yr+uLK2tVbceJ59inm4Dw+ghO/7CWzAS9iEOjBtoL3VPmkns79zJPcwV06l2kzG3IWJK1f9C1UROaI=</latexit>

cc

Z T2

Z T2
T1

c dT / T = - dV / V をこの範囲で別々に積分する

T1

1dT =
Td ln TdT =dTT2c [ln T ]T1 =c(ln T2ln T1 ) =

Z V2

V1
Z V2
V1

1dVV
d ln VdVdV

V2[ln V ]V1(ln V2ln V1 )
✓ ◆c
T2
V1T2
V1c ln= ln , ln=T1
V2
T1V2✓ ◆cV1T2= ln , V1 T1c = V2 T2c
T1V2

Adiabatic changes - Poisson equations

断熱変化

ポアソンの関係式<latexit sha1_base64="XHdq7nTOmwu9irxNJAJZ1OwxHGQ=">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</latexit>

cc
V1 T1 = V2 T2 ,c

V T = const, (V T )P V = nRT, T = P V /(nR), VPV

c+1cc 1/c

1/c=V1/c

T = const

P V = const= const
CV c + 1
CV + nRCP,==⌘c=nRc
CVCV比熱比（heat capacity ratio)
P V = const可逆的断熱膨張：仕事T: TH →TL
V: V2→V3
P: P2→P3

V2 < V3P2V2 = nRTHP3V3 = nRTL<latexit sha1_base64="s074OK+AtLh1JfdFX4CrlEAiDkI=">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</latexit>

PV= const = P2 V2 = P3 V3Z V3Z V3P dV = P2 V2V
W =V2=====P2 V 21V+1dV

V2V3+1=V2P2 V 21(V3+1+1V21( P3 V3 V3 +1 + P2 V2 V2 +1 )+11nR
( P3 V 3 + P2 V 2 ) =( TL + TH )+1+1nRnR( TL + TH ) = CV
( TL + TH )CP /CV + 1(CP CV )CP CV
( TL + TH ) = CV (TL TH ) < 0CV
CP CV+1)