در فهرست زیر معادلات متداول و پر کاربرد ترمودینامیک آورده شده‌است:
متغیرهای متداول
W
{\displaystyle W}
کار انجام شوده به وسیلهٔ سیستم بر روی محیط
Q
{\displaystyle Q}
گرما انتقال یافته از محیط به سیستم
δw
Infinitesimal amount of Work
δq
Infinitesimal amount of Heat
S
=
k
B
(
ln
Ω
)
{\displaystyle ~S=k_{B}(\ln \Omega )~}
k
B
{\displaystyle k_{B}}
ثابت بولتزمن ، and
Ω
{\displaystyle \Omega }
ریزحالت in the فضای فاز .
d
S
=
δ
Q
T
{\displaystyle ~dS={\frac {\delta Q}{T}}~}
خواص کوانتومی
ویرایش
U
=
N
k
B
T
2
(
∂
ln
Z
∂
T
)
V
{\displaystyle ~U=Nk_{B}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}~}
S
=
U
T
+
N
∗
S
=
U
T
+
N
k
B
ln
Z
−
N
k
ln
N
+
N
k
{\displaystyle ~S={\frac {U}{T}}+N*~S={\frac {U}{T}}+Nk_{B}\ln Z-Nk\ln N+Nk~}
where N is number of particles, Z is the partition function , h is ثابت پلانک ، I is ممان اینرسی ، Zt is Ztranslation , Zv is Zvibration , Zr is Zrotation
Z
t
=
(
2
π
m
k
B
T
)
3
2
V
h
3
{\displaystyle ~Z_{t}={\frac {(2\pi mk_{B}T)^{\frac {3}{2}}V}{h^{3}}}~}
Z
v
=
1
1
−
e
−
h
ω
2
π
k
B
T
{\displaystyle ~Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{B}T}}}}~}
Z
r
=
2
I
k
B
T
σ
(
h
2
π
)
2
{\displaystyle ~Z_{r}={\frac {2Ik_{B}T}{\sigma ({\frac {h}{2\pi }})^{2}}}~}
d
Q
=
C
p
d
T
+
l
v
d
v
=
d
U
+
P
d
V
=
T
d
S
{\displaystyle ~dQ=C_{p}dT+l_{v}d_{v}=dU+PdV=TdS~}
C
p
=
(
∂
Q
r
e
v
∂
T
)
p
=
(
∂
U
∂
T
)
p
+
p
(
∂
V
∂
T
)
p
=
(
∂
H
∂
T
)
p
=
T
(
∂
S
∂
T
)
p
{\displaystyle ~C_{p}=\left({\partial Q_{rev} \over \partial T}\right)_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}~}
C
V
=
(
∂
Q
r
e
v
∂
T
)
V
=
(
∂
U
∂
T
)
V
=
T
(
∂
S
∂
T
)
V
{\displaystyle ~C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}~}
نام
نماد
فرمول
متغیرهای طبیعی
انرژی درونی
U
{\displaystyle U}
∫
(
T
d
S
−
p
d
V
+
∑
i
μ
i
d
N
i
)
{\displaystyle \int (TdS-pdV+\sum _{i}\mu _{i}dN_{i})}
S
,
V
,
{
N
i
}
{\displaystyle S,V,\{N_{i}\}}
انرژی آزاد هلمولتز
F
{\displaystyle F}
U
−
T
S
{\displaystyle U-TS}
T
,
V
,
{
N
i
}
{\displaystyle T,V,\{N_{i}\}}
آنتالپی
H
{\displaystyle H}
U
+
p
V
{\displaystyle U+pV}
S
,
p
,
{
N
i
}
{\displaystyle S,p,\{N_{i}\}}
انرژی آزاد گیبس
G
{\displaystyle G}
U
+
p
V
−
T
S
{\displaystyle U+pV-TS}
T
,
p
,
{
N
i
}
{\displaystyle T,p,\{N_{i}\}}
پتانسیل لاندو (پتانسیل بزرگ)
Ω
{\displaystyle \Omega }
Φ
G
{\displaystyle \Phi _{G}}
U
−
T
S
−
{\displaystyle U-TS-}
∑
i
{\displaystyle \sum _{i}\,}
μ
i
N
i
{\displaystyle \mu _{i}N_{i}}
T
,
V
,
{
μ
i
}
{\displaystyle T,V,\{\mu _{i}\}}
See also:
K
T
=
−
1
V
(
∂
V
∂
p
)
T
,
N
{\displaystyle ~K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}~}
برخی روابط دیگر
ویرایش
(
∂
S
∂
U
)
V
,
N
=
1
T
{\displaystyle ~\left({\partial S \over \partial U}\right)_{V,N}={1 \over T}~}
(
∂
S
∂
V
)
N
,
U
=
p
T
{\displaystyle ~\left({\partial S \over \partial V}\right)_{N,U}={p \over T}~}
(
∂
S
∂
N
)
V
,
U
=
−
μ
T
{\displaystyle ~\left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}~}
(
∂
T
∂
S
)
V
=
T
C
V
{\displaystyle ~\left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}~}
(
∂
T
∂
S
)
p
=
T
C
p
{\displaystyle ~\left({\partial T \over \partial S}\right)_{p}={T \over C_{p}}~}
−
(
∂
p
∂
V
)
T
=
1
V
K
T
{\displaystyle ~-\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}~}
Quantity
General Equation
Isobaricp = ۰
IsochoricV = ۰
IsothermalT = ۰
Adiabatic
Q
=
0
{\displaystyle Q=0}
کار
W
{\displaystyle W}
δ
W
=
p
d
V
{\displaystyle \delta W=pdV\;}
p
Δ
V
{\displaystyle p\Delta V\;}
0
{\displaystyle 0\;}
n
R
T
ln
V
2
V
1
{\displaystyle nRT\ln {\frac {V_{2}}{V_{1}}}\;}
P
V
γ
(
V
f
1
−
γ
−
V
i
1
−
γ
)
1
−
γ
{\displaystyle {\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}}
[۱]
C
V
(
T
1
−
T
2
)
{\displaystyle C_{V}\left(T_{1}-T_{2}\right)}
ظرفیت گرماییC
(as for real gas)
C
p
=
5
2
n
R
{\displaystyle C_{p}={\frac {5}{2}}nR\;}
C
V
=
3
2
n
R
{\displaystyle C_{V}={\frac {3}{2}}nR\;}
انرژی درونیU
Δ
U
=
C
v
Δ
T
{\displaystyle \Delta U=C_{v}\Delta T\;}
Q
+
W
{\displaystyle Q+W\;}
Q
p
−
p
Δ
V
{\displaystyle Q_{p}-p\Delta V\;}
Q
{\displaystyle Q\;}
C
V
(
T
2
−
T
1
)
{\displaystyle C_{V}\left(T_{2}-T_{1}\right)\;}
0
{\displaystyle 0\;}
Q
=
W
{\displaystyle Q=W\;}
−
W
{\displaystyle -W\;}
C
V
(
T
2
−
T
1
)
{\displaystyle C_{V}\left(T_{2}-T_{1}\right)\;}
آنتالپیH
H
=
U
+
p
V
{\displaystyle H=U+pV\;}
C
p
(
T
2
−
T
1
)
{\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}
Q
V
+
V
Δ
p
{\displaystyle Q_{V}+V\Delta p\;}
0
{\displaystyle 0\;}
C
p
(
T
2
−
T
1
)
{\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}
آنتروپیS
Δ
S
=
C
v
ln
T
2
T
1
+
R
ln
V
2
V
1
{\displaystyle \Delta S=C_{v}\ln {T_{2} \over T_{1}}+R\ln {V_{2} \over V_{1}}}
Δ
S
=
C
p
ln
T
2
T
1
−
R
ln
p
2
p
1
{\displaystyle \Delta S=C_{p}\ln {T_{2} \over T_{1}}-R\ln {p_{2} \over p_{1}}}
[۲]
C
p
ln
T
2
T
1
{\displaystyle C_{p}\ln {\frac {T_{2}}{T_{1}}}\;}
C
V
ln
T
2
T
1
{\displaystyle C_{V}\ln {\frac {T_{2}}{T_{1}}}\;}
n
R
ln
V
2
V
1
{\displaystyle nR\ln {\frac {V_{2}}{V_{1}}}\;}
Q
T
{\displaystyle {\frac {Q}{T}}\;}
C
p
ln
V
2
V
1
+
C
V
ln
p
2
p
1
=
0
{\displaystyle C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;}
Constant
{\displaystyle \;}
V
T
{\displaystyle {\frac {V}{T}}\;}
p
T
{\displaystyle {\frac {p}{T}}\;}
p
V
{\displaystyle pV\;}
p
V
γ
{\displaystyle pV^{\gamma }\;}
دیگر معادلات سودمند
ویرایش
Δ
U
=
Q
−
W
=
Q
−
∫
p
e
x
t
d
V
=
Q
−
p
e
x
t
Δ
V
{\displaystyle \Delta U=Q-W=Q-\int p_{ext}dV=Q-p_{ext}\Delta V}
H
=
U
+
p
V
{\displaystyle H=U+pV\,\!}
A
=
U
−
T
S
{\displaystyle A=U-TS\,\!}
G
=
H
−
T
S
=
∑
i
μ
i
N
i
{\displaystyle G=H-TS=\sum _{i}\mu _{i}N_{i}\,\!}
d
U
(
S
,
V
,
n
i
)
=
T
d
S
−
p
d
V
+
∑
i
μ
i
d
N
i
{\displaystyle dU\left(S,V,{n_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}}
d
H
(
S
,
p
,
n
i
)
=
T
d
S
+
V
d
p
+
∑
i
μ
i
d
N
i
{\displaystyle dH\left(S,p,n_{i}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}}
d
A
(
T
,
V
,
n
i
)
=
−
S
d
T
−
p
d
V
+
∑
i
μ
i
d
N
i
{\displaystyle dA\left(T,V,n_{i}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}}
d
G
(
T
,
p
,
n
i
)
=
−
S
d
T
+
V
d
p
+
∑
i
μ
i
d
N
i
{\displaystyle dG\left(T,p,n_{i}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}}
μ
J
T
=
(
∂
T
∂
p
)
H
{\displaystyle \mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}}
κ
T
=
−
1
V
(
∂
V
∂
p
)
T
{\displaystyle \kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T}}
α
p
=
1
V
(
∂
V
∂
T
)
p
{\displaystyle \alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}}
(
∂
H
∂
p
)
T
=
V
−
T
(
∂
V
∂
T
)
p
{\displaystyle \left({\frac {\partial H}{\partial p}}\right)_{T}=V-T\left({\frac {\partial V}{\partial T}}\right)_{p}}
(
∂
U
∂
V
)
T
=
T
(
∂
p
∂
T
)
V
−
p
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}
H
=
−
T
2
(
∂
(
G
/
T
)
∂
T
)
p
{\displaystyle H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}}
U
=
−
T
2
(
∂
(
A
/
T
)
∂
T
)
V
{\displaystyle U=-T^{2}\left({\frac {\partial \left(A/T\right)}{\partial T}}\right)_{V}}
نمونه ای از کاربرد روش بالا:
(
∂
T
∂
p
)
H
=
−
1
C
p
(
∂
H
∂
p
)
T
{\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}
(
∂
T
∂
p
)
H
(
∂
p
∂
H
)
T
(
∂
H
∂
T
)
p
=
−
1
{\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}\left({\frac {\partial p}{\partial H}}\right)_{T}\left({\frac {\partial H}{\partial T}}\right)_{p}=-1}
(
∂
T
∂
p
)
H
=
−
(
∂
H
∂
p
)
T
(
∂
T
∂
H
)
p
{\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}=-\left({\frac {\partial H}{\partial p}}\right)_{T}\left({\frac {\partial T}{\partial H}}\right)_{p}}
=
−
1
(
∂
H
∂
T
)
p
(
∂
H
∂
p
)
T
{\displaystyle ={\frac {-1}{\left({\frac {\partial H}{\partial T}}\right)_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}
C
p
=
(
∂
H
∂
T
)
p
{\displaystyle C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}}
⇒
(
∂
T
∂
p
)
H
=
−
1
C
p
(
∂
H
∂
p
)
T
{\displaystyle \Rightarrow \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}
نمونه‌های دیگر:
C
V
=
T
(
∂
S
∂
T
)
V
{\displaystyle C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}
‴
U
=
Q
−
W
‴
{\displaystyle '''U=Q-W\,\!'''}
d
U
=
δ
Q
r
e
v
−
δ
W
r
e
v
;
d
S
=
δ
Q
r
e
v
T
,
δ
W
r
e
v
=
p
d
V
{\displaystyle dU=\delta Q_{rev}-\delta W_{rev};dS={\frac {\delta Q_{rev}}{T}},\delta W_{rev}=pdV\,\!}
=
T
d
S
−
p
d
V
{\displaystyle =TdS-pdV\,\!}
(
∂
U
∂
T
)
V
=
T
(
∂
S
∂
T
)
V
−
p
(
∂
V
∂
T
)
V
;
C
V
=
(
∂
U
∂
T
)
V
{\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}-p\left({\frac {\partial V}{\partial T}}\right)_{V};C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}}
⇒
C
V
=
T
(
∂
S
∂
T
)
V
{\displaystyle \Rightarrow C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}
جستارهای وابسته
ویرایش
Atkins, Peter and de Paula, Julio Physical Chemistry , 7th edition, W.H. Freeman and Company, 2002 ISBN 0-7167-3539-3 ].
Chapters 1 - 10, Part 1: Equilibrium . Bridgman, P.W. , Phys. Rev. , 3, 273 (1914).
Landsberg, Peter T. Thermodynamics and Statistical Mechanics . New York: Dover Publications, Inc. , 1990. (reprinted from Oxford University Press, 1978) .
Lewis, G.N. , and Randall, M. , "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.
Reichl, L.E. , "A Modern Course in Statistical Physics", 2nd edition, New York: John Wiley & Sons, 1998.
Schroeder, Daniel V. Thermal Physics . San Francisco: Addison Wesley Longman, 2000 ISBN 0-201-38027-7 ].
Silbey, Robert J. , et al. Physical Chemistry . 4th ed. New Jersey: Wiley, 2004.
Callen, Herbert B. (1985). "Thermodynamics and an Introduction to Themostatistics", 2nd Ed. , New York: John Wiley & Sons.
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