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U =q+w
H = U + PV
G = H TS
A = U TS
dS =
dU = dq + dw = TdS PdV
dH = dU + PdV + VdP = TdS PdV + PdV + VdP = TdS + VdP
dG = dH TdS SdT = TdS + VdP TdS SdT = VdP SdT
dA = dU TdS SdT = TdS PdV TdS SdT = PdV SdT
dq
, dw = PdV
T
dU = dq + dw = TdS PdV
dU dU = TdS =T
dS v
dU dU = PdV = P
dV S
dH = TdS + VdP
dH =T
dS P
dH =V
dP S
dG = VdP SdT
dG = S
dT P
dG =V
dP T
dA = PdV SdT
dA = P
dV T
dA = S
dT V
Z = f ( x, y ) f (x, y) Z Z dZ = dx + dy
x y
y x
Z Z = y x y x x y x y
U U dU = TdS PdV = dS + dV
S V
V S
U U = V S V S S V S V
dU =T
dS v
T P = V S
S V
dU = P
dV S
H H dH = TdS + VdP = dS + dP
S P
P S
H H = P S P S S P S P
dH =T
dS P
T V = P S S P
dH =V
dP S
G G dG = VdP SdT = dP + dT
P T
T P
G G = T P T P P T P T
dG =V
dP T
V S = T P
P T
dG = S
dT P
A A dA = PdV SdT = dV + dT
V T
T V
A A = T V T V V T V T
dA = P
dV T
dA = S
dT V
P S = T V V T
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P T
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ƒ€2H…CP, CV† ƒP-­‐V-­‐T J
J!
ƒ
Gq{€s„
ƒq‚w€|„ ƒq‚z|„ Rjf€2Hd
⎛ ∂H ⎞
CP = ⎜
,
⎝ ∂T ⎟⎠ P
⎛ ∂U ⎞
CV = ⎜
⎝ ∂T ⎟⎠ V
q‚z|„o]^>Y`
dH = TdS + VdP jl‡ P = _d‡dH = TdS
⎛ ∂H ⎞
⎛ ∂S ⎞
CP = ⎜
=T⎜
⎟
⎝ ∂T ⎠ P
⎝ ∂T ⎟⎠ P
dU = TdS − PdV jl‡ V = _d‡dU = TdS
⎛ ∂U ⎞
⎛ ∂S ⎞
CV = ⎜
=T⎜
⎟
⎝ ∂T ⎠ V
⎝ ∂T ⎟⎠ V
q‚z|„ S o V, T cˆ(J( S = f(V, T) `Ym`
C
⎛ ∂S ⎞
⎛ ∂S ⎞
⎛ ∂P ⎞
dS = ⎜
dV + ⎜
dT = ⎜
dV + V dT
⎟
⎟
⎟
⎝ ∂V ⎠ T
⎝ ∂T ⎠ V
⎝ ∂T ⎠ V
T
Maxwell cJ!jl
g[‡q‚z|„No S = f(P, T) `Ym`
C
⎛ ∂S ⎞
⎛ ∂S ⎞
⎛ ∂V ⎞
dS = ⎜
dP + ⎜
dT = − ⎜
dP + P dT
⎟
⎟
⎟
⎝ ∂P ⎠ T
⎝ ∂T ⎠ P
⎝ ∂T ⎠ P
T
Maxwell cJ!jl
WnT‡P-­‐V-­‐T J`q‚z|„o;V!_PmO
U H S Gq{€s„ U d dU = TdS − PdV jl‡U = f(V, T) `X^‡_/i[NdS cJ!oYne
⎫
⎫
⎧⎛ ∂P ⎞
⎧ ⎛ ∂P ⎞
CV
dU = T ⎨⎜
dT ⎬ − PdV = ⎨T ⎜
⎟⎠ dV +
⎟⎠ − P ⎬ dV + CV dT
⎝
⎝
T
⎩ ∂T V
⎭
⎩ ∂T V
⎭
g[‡U = f(P, T) `X^‡dS `cJ!oYne‡
⎫
⎧ ⎛ ∂V ⎞
C
⎛ ∂V ⎞
dU = T ⎨− ⎜
dP + P dT ⎬ − PdV = −T ⎜
dP + CP dT − PdV
⎟
⎝ ∂T ⎟⎠ P
T
⎩ ⎝ ∂T ⎠ P
⎭
,b‡q‚w€|„ H d dH = TdS + VdP jl‡H = f(V, T) `X^‡NdS cJ!oYne
⎫
⎧⎛ ∂P ⎞
C
⎛ ∂P ⎞
dH = T ⎨⎜
dV + V dT ⎬ + VdP = T ⎜
dV + CV dT + VdP
⎟
⎝ ∂T ⎟⎠ V
T
⎩⎝ ∂T ⎠ V
⎭
g[‡H = f(P, T) `X^‡dS `cJ!oYne‡
⎫
⎤
⎧ ⎛ ∂V ⎞
⎡ ⎛ ∂V ⎞
C
dH = T ⎨− ⎜
dP + P dT ⎬ + VdP = ⎢ −T ⎜
+ V ⎥ dP + CP dT
⎟
⎟
T
⎣ ⎝ ∂T ⎠ P
⎦
⎩ ⎝ ∂T ⎠ P
⎭
WW_‡CP ` CV `cJo/i^hmO
⎛ ∂U ⎞
⎛ ∂S ⎞
CV = ⎜
=T⎜
⎝ ∂T ⎟⎠ V
⎝ ∂T ⎟⎠ V
C
⎛ ∂S ⎞
⎛ ∂S ⎞
⎛ ∂V ⎞
dS = ⎜
dP + ⎜
dT = − ⎜
dP + P dT
⎟
⎟
⎟
⎝ ∂P ⎠ T
⎝ ∂T ⎠ P
⎝ ∂T ⎠ P
T
C
C
⎛ ∂S ⎞
⎛ ∂V ⎞ ⎛ ∂P ⎞
∴⎜
= −⎜
+ P= V
⎟
⎟
⎜
⎟
⎝ ∂T ⎠ V
⎝ ∂T ⎠ P ⎝ ∂T ⎠ V T
T
⎛ ∂V ⎞ ⎛ ∂P ⎞
∴ CP − CV = T ⎜
⎝ ∂T ⎟⎠ P ⎜⎝ ∂T ⎟⎠ V
4%.c‡PV = RT jl‡
R ⎛ ∂P ⎞
R
⎛ ∂V ⎞
⎜⎝
⎟⎠ = , ⎜⎝
⎟⎠ =
∂T P P
∂T V V
2
⎛ R ⎞ ⎛ R ⎞ TR
∴ CP − CV = T ⋅ ⎜ ⎟ ⋅ ⎜ ⎟ =
=R
⎝ P ⎠ ⎝ V ⎠ RT
Gibbs – Helmholtz c!
G = H − TS
⎛ ∂G ⎞
S = −⎜
⎝ ∂T ⎟⎠ P
⎛ ∂G ⎞
∴ G = H +T ⎜
⎝ ∂T ⎟⎠ P
∴ TdG − GdT = −HdT
P : const
Eo T2 _m`
TdG − GdT
dT
= −H 2
2
T
T
⎛ G⎞
d⎜ ⎟
⎝T⎠
P : const
⎛ 1⎞
Hd ⎜ ⎟
⎝T ⎠
⎡ ∂( G T ) ⎤
∴ ⎢
⎥ =H
∂
1
T
(
)
⎣
⎦P
Gibbs – Helmholtz c!

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