جدول معادلات ترمودینامیکی

از ویکی‌پدیا، دانشنامهٔ آزاد
پرش به: ناوبری، جستجو

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

متغیر ها[ویرایش]

نوشتار اصلی: فهرست خواص ترمودینامیکی


معادله ها[ویرایش]

آنتروپی[ویرایش]

خواص کوانتومی[ویرایش]

  • ~ 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, Zt is Ztranslation, Zv is Zvibration, Zr is Zrotation

  • ~ 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:

Quasi-static and reversible processes[ویرایش]

  • ~ dQ=C_p dT+l_v d_v 
=dU+PdV
=TdS~

ظرفیت گرمایی در فشار ثابت[ویرایش]

  • ~ 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[ویرایش]

  • ~ 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[ویرایش]

Name Symbol Formula Natural variables
انرژی درونی U \int ( T dS - p dV + \sum_i \mu_i dN_i ) S, V, \{N_i\}
انرژی آزاد هلمولتز F U-TS T, V, \{N_i\}
آنتالپی H U+pV S, p, \{N_i\}
انرژی آزاد گیبس G U+pV-TS T, p, \{N_i\}
Landau Potential (Grand potential) \Omega, \Phi_{G} U - T S -\sum_i\,\mu_i N_i T, V, \{\mu_i\}

See also:

ضریب تراکم‌پذیری همدما at constant temperature[ویرایش]

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

برخی روابط دیگر[ویرایش]

  • ~ \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[ویرایش]

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} [۱] = 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}[۲]
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[ویرایش]

  • \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[ویرایش]

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[ویرایش]

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[ویرایش]

منابع[ویرایش]

  1. Adiabatic Processes
  2. Keenan, Thermodynamics, Wiley, New York, 1947
  • 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.