جدول معادلات ترمودینامیکی

در فهرست زیر معادلات متداول و پر کاربرد ترمودینامیک آورده شده است:

متغیر ها

نوشتار اصلی: فهرست خواص ترمودینامیکی

Conjugate variables

p	فشار
V	حجم (ترمودینامیک)
T	دما
S	انتروپی
μ	پتانسیل شیمیایی
N	تعداد ذره

پتانسیل ترمودینامیکی

U	انرژی درونی
A	انرژی آزاد هلمولتز
H	آنتالپی
G	انرژی آزاد گیبس

Material properties

ρ	چگالی
CV	ظرفیت گرمایی (حجم ثابت)
CP	ظرفیت گرمایی (فشار ثابت)
βT	Isothermal ضریب تراکم‌پذیری همدما
βS	Adiabatic ضریب تراکم‌پذیری همدما
α	ضریب انبساط گرمایی

ثابت‌ها

kB	ثابت بولتزمن
R	ثابت عمومی گازها

متغیرهای متداول

$W$	کار انجام شوده به وسیلهٔ سیستم بر روی محیط
$Q$	گرما انتقال یافته از محیط به سیستم
δw	Infinitesimal amount of Work
δq	Infinitesimal amount of Heat

معادله ها

آنتروپی

• $~ S = k_B (\ln \Omega) ~$, where $k_B$ is the ثابت بولتزمن, and $\Omega$ denotes the volume of macrostate in the فضای فاز.
• $~ dS = \frac{\delta Q}{T} ~$, برای سیستم برگشت پذیر

خواص کوانتومی

• $~ U = N k_B T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V ~$

• $~ S = \frac{U}{T} + N * ~ S = \frac{U}{T} + N k_B \ln Z - N k \ln N + Nk ~$ 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

• $~ Z_t = \frac{(2 \pi m k_B T)^\frac{3}{2} V}{h^3} ~$

• $~ Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_B T}} ~$

• $~ Z_r = \frac{2 I k_B T}{\sigma (\frac{h}{2 \pi})^2} ~$

where:

• $~ \sigma = 1 ~$, heteronuclear
• $~ \sigma = 2 ~$, homonuclear

Quasi-static and reversible processes

• $~ dQ=C_p dT+l_v d_v =dU+PdV =TdS~$

ظرفیت گرمایی در فشار ثابت

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

ظرفیت گرمایی در حجم ثابت

• $~ C_V = \left ( {\partial Q_{rev} \over \partial T} \right )_V = \left ( {\partial U \over \partial T} \right )_V = T \left ( {\partial S \over \partial T} \right )_V ~$

پتانسیل ترمودینامیکیs and related concepts

See also:

• روابط ماکسول
• روابط اساسی ترمودینامیک

ضریب تراکم‌پذیری همدما at constant temperature

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

برخی روابط دیگر

• $~ \left ( {\partial S\over \partial U} \right )_{V,N} = { 1\over T } ~$
• $~ \left ( {\partial S\over \partial V} \right )_{N,U} = { p\over T } ~$
• $~ \left ( {\partial S\over \partial N} \right )_{V,U} = - { \mu \over T } ~$
• $~ \left ( {\partial T\over \partial S} \right )_V = { T \over C_V } ~$
• $~ \left ( {\partial T\over \partial S} \right )_p = { T \over C_p } ~$
• $~ -\left ( {\partial p\over \partial V} \right )_T = { 1 \over {VK_T} } ~$

Equation Table for an گاز ایده‌آل

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic $Q=0$
کار $W$	$\delta W = p dV\;$	$p\Delta V\;$	$0\;$	$nRT\ln\frac{V_2}{V_1}\;$	$\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma}$ = $C_V \left(T_1 - T_2 \right)$
ظرفیت گرمایی C	(as for real gas)	$C_p = \frac{5}{2}nR\;$ (for monatomic ideal gas)	$C_V = \frac{3}{2}nR \;$ (for monatomic ideal gas)
انرژی درونی ΔU	$\Delta U = C_v \Delta T\;$	$Q - W\;$ $Q_p - p\Delta V\;$	$Q\;$ $C_V\left ( T_2-T_1 \right )\;$	$0\;$ $Q=W\;$	$-W\;$ $C_V\left ( T_2-T_1 \right )\;$
آنتالپی ΔH	$H=U+pV\;$	$C_p\left ( T_2-T_1 \right )\;$	$Q_V+V\Delta p\;$	$0\;$	$C_p\left ( T_2-T_1 \right )\;$
آنتروپی ΔS	$\Delta S = C_v \ln{T_2 \over T_1} + R \ln{V_2 \over V_1}$ $\Delta S = C_p \ln{T_2 \over T_1} - R \ln{p_2 \over p_1}$	$C_p\ln\frac{T_2}{T_1}\;$	$C_V\ln\frac{T_2}{T_1}\;$	$nR\ln\frac{V_2}{V_1}\;$ $\frac{Q}{T}\;$	$C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;$
Constant	$\;$	$\frac{V}{T}\;$	$\frac{p}{T}\;$	$p V\;$	$p V^\gamma\;$

Other useful identities

• $\Delta U = Q - W = Q - \int p_{ext} dV = Q - p_{ext}\Delta V$

• $H = U + pV \,\!$
• $A = U - TS \,\!$

• $G = H - TS = \sum_{i} \mu_{i} N_{i} \,\!$
• $dU\left(S,V,{n_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i$
• $dH\left(S,p,n_{i}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}$
• $dA\left(T,V,n_{i}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}$
• $dG\left(T,p,n_{i}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}$

• $\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H$
• $\kappa_{T} = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T$
• $\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$

• $\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_p$
• $\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$
• $H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p$
• $U = -T^2\left(\frac{\partial \left(A/T\right)}{\partial T}\right)_V$

Proof #1

An example using the above methods is:

$\left(\frac{\partial T}{\partial p}\right)_H = -\frac{1}{C_p} \left(\frac{\partial H}{\partial p}\right)_T$

$\left(\frac{\partial T}{\partial p}\right)_H \left(\frac{\partial p}{\partial H}\right)_T \left(\frac{\partial H}{\partial T}\right)_p = -1$

$\left(\frac{\partial T}{\partial p}\right)_H = -\left(\frac{\partial H}{\partial p}\right)_T \left(\frac{\partial T}{\partial H}\right)_p$

$= \frac{-1}{\left(\frac{\partial H}{\partial T}\right)_p} \left(\frac{\partial H}{\partial p}\right)_T$ ; $C_p = \left(\frac{\partial H}{\partial T}\right)_p$

$\Rightarrow \left(\frac{\partial T}{\partial p}\right)_H = -\frac{1}{C_p} \left(\frac{\partial H}{\partial p}\right)_T$

Proof #2

نمونه های دیگر:

$C_V = T\left(\frac{\partial S}{\partial T}\right)_V$

$'''U = Q - W \,\!'''$

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

$= TdS-pdV \,\!$

$\left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V - p\left(\frac{\partial V}{\partial T}\right)_V ; C_V = \left(\frac{\partial U}{\partial T}\right)_V$

$\Rightarrow C_V = T\left(\frac{\partial S}{\partial T}\right)_V$

جستارهای وابسته

• معادله آنتوان
• عدد بیجان
• Bowen ratio
• Bridgman's equations
• معادله کلازیوس-کلاپیرون
• Departure functions
• Duhem–Margules equation
• Ehrenfest equations
• Gibbs–Helmholtz equation
• قانون فازی گیبس
• Kopp's law
• Kopp–Neumann law
• Noro–Frenkel law of corresponding states
• Onsager reciprocal relations
• عدد استفان
• Triple product rule

منابع

• 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.