جدول معادلات ترمودینامیکی

کمیت (نام رایج)	نماد (رایج)	یکای SI	بعد
تعداد مولکول‌ها	N	بدون بعد	بدون بعد
تعداد مول‌ها	n	mol	[N]
دما	T	K	[Θ]
گرما	Q, q	J	${\displaystyle [M][L]^{2}[T]^{-2}}$
گرمای نهان	${\displaystyle Q_{L}}$	J	${\displaystyle [M][L]^{2}[T]^{-2}}$

• ${\displaystyle ~S=k_{B}(\ln \Omega )~}$، where ${\displaystyle k_{B}}$ is the ثابت بولتزمن، and ${\displaystyle \Omega }$ denotes the volume of ریزحالت in the فضای فاز.
• ${\displaystyle ~dS={\frac {\delta Q}{T}}~}$، برای سیستم برگشت‌پذیر

• ${\displaystyle ~U=Nk_{B}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}~}$
• ${\displaystyle ~S={\frac {U}{T}}+N*~S={\frac {U}{T}}+Nk_{B}\ln Z-Nk\ln N+Nk~}$ Indistinguishable Particles

where N is number of particles, Z is the partition function, h is ثابت پلانک، I is ممان اینرسی، Z_t is Z_translation, Z_v is Z_vibration, Z_r is Z_rotation

• ${\displaystyle ~Z_{t}={\frac {(2\pi mk_{B}T)^{\frac {3}{2}}V}{h^{3}}}~}$
• ${\displaystyle ~Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{B}T}}}}~}$
• ${\displaystyle ~Z_{r}={\frac {2Ik_{B}T}{\sigma ({\frac {h}{2\pi }})^{2}}}~}$

where:

• ${\displaystyle ~\sigma =1~}$، مولکول ناجورهسته
• ${\displaystyle ~\sigma =2~}$، مولکول جورهسته

• ${\displaystyle ~dQ=C_{p}dT+l_{v}d_{v}=dU+PdV=TdS~}$

• ${\displaystyle ~C_{p}=\left({\partial Q_{rev} \over \partial T}\right)_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}~}$

• ${\displaystyle ~C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}~}$

نام	نماد	فرمول	متغیرهای طبیعی
انرژی درونی	${\displaystyle U}$	${\displaystyle \int (TdS-pdV+\sum _{i}\mu _{i}dN_{i})}$	${\displaystyle S,V,\{N_{i}\}}$
انرژی آزاد هلمولتز	${\displaystyle F}$	${\displaystyle U-TS}$	${\displaystyle T,V,\{N_{i}\}}$
آنتالپی	${\displaystyle H}$	${\displaystyle U+pV}$	${\displaystyle S,p,\{N_{i}\}}$
انرژی آزاد گیبس	${\displaystyle G}$	${\displaystyle U+pV-TS}$	${\displaystyle T,p,\{N_{i}\}}$
پتانسیل لاندو (پتانسیل بزرگ)	${\displaystyle \Omega }$, ${\displaystyle \Phi _{G}}$	${\displaystyle U-TS-}$${\displaystyle \sum _{i}\,}$${\displaystyle \mu _{i}N_{i}}$	${\displaystyle T,V,\{\mu _{i}\}}$

See also:

• روابط ماکسول
• روابط اساسی ترمودینامیک

• ${\displaystyle ~K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}~}$

• ${\displaystyle ~\left({\partial S \over \partial U}\right)_{V,N}={1 \over T}~}$
• ${\displaystyle ~\left({\partial S \over \partial V}\right)_{N,U}={p \over T}~}$
• ${\displaystyle ~\left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}~}$
• ${\displaystyle ~\left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}~}$
• ${\displaystyle ~\left({\partial T \over \partial S}\right)_{p}={T \over C_{p}}~}$
• ${\displaystyle ~-\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}~}$

نمونه ای از کاربرد روش بالا:

${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}$

${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}\left({\frac {\partial p}{\partial H}}\right)_{T}\left({\frac {\partial H}{\partial T}}\right)_{p}=-1}$

${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{H}=-\left({\frac {\partial H}{\partial p}}\right)_{T}\left({\frac {\partial T}{\partial H}}\right)_{p}}$

${\displaystyle ={\frac {-1}{\left({\frac {\partial H}{\partial T}}\right)_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}$ ; ${\displaystyle C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}}$

${\displaystyle \Rightarrow \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}}$

نمونه‌های دیگر:

${\displaystyle C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}$

${\displaystyle '''U=Q-W\,\!'''}$

${\displaystyle dU=\delta Q_{rev}-\delta W_{rev};dS={\frac {\delta Q_{rev}}{T}},\delta W_{rev}=pdV\,\!}$

${\displaystyle =TdS-pdV\,\!}$

${\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}-p\left({\frac {\partial V}{\partial T}}\right)_{V};C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}}$

${\displaystyle \Rightarrow C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}$