Thermodynamic equations

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In thermodynamics, there are a large number of equations relating the various thermodynamic quantities. Some of the most common thermodynamic quantites are:

The following equations are classified by subject.

[edit] First law of thermodynamics

<math>~ dU=\delta q-\delta w ~</math>

Note that the symbol <math>\delta</math> represents the fact that because q and w are not state functions, <math>\delta q</math> and <math>\delta w</math> are inexact differentials.

In some fields such as physical chemistry, positive work is conventionally considered work done on the system rather than by the system, and the law is expressed as <math>dU=\delta q+\delta w</math>.

[edit] Entropy

<math>~ S = k (\ln \Omega) ~</math>
<math>~ \Delta S = \frac{\Delta Q}{T} ~</math>

[edit] Quasi-static and reversible process

<math>~ dQ=C_V dT+l_v d_v 

=dU+PdV =TdS~ </math>

[edit] Heat capacity at constant pressure

<math>~ C_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 ~</math>

[edit] Heat capacity at constant volume

<math>~ C_V=\left ( {\partial U\over \partial T} \right )_V

= T \left ( {\partial S\over \partial T} \right )_V ~</math>

[edit] Helmholtz free energy

<math>~ A \equiv U-TS = \mu n - PV ~</math>

[edit] Gibbs free energy

<math>~ G \equiv U-TS+PV = \mu n ~</math>

[edit] Enthalpy

<math>~ H \equiv U+PV = \mu n + TS~</math>

[edit] Maxwell relations

<math>~ \left ( {\partial T\over \partial V} \right )_{S,n} 

<math>~ \left ( {\partial T\over \partial P} \right )_{S,n} 

<math>~ \left ( {\partial T\over \partial V} \right )_{P,n} 

<math>~ \left ( {\partial T\over \partial P} \right )_{V,n} 

[edit] Incremental processes

<math>~ dU = T\,dS-P\,dV + \mu\,dn ~</math>
<math>~ dA = -S\,dT-P\,dV + \mu\,dn ~</math>
<math>~ dG = -S\,dT+V\,dP + \mu\,dn = \mu\,dn +n\,d\mu ~</math>
<math>~ dH = T\,dS+V\,dP + \mu\,dn ~</math>

[edit] Compressibility at constant temperature

<math>~ K_T = -{ 1\over V } \left ( {\partial V\over \partial P} \right )_{T,n} ~</math>

[edit] More relations

<math>~ \left ( {\partial S\over \partial U} \right )_{V,n} 

<math>~ \left ( {\partial S\over \partial V} \right )_{n,U} 

<math>~ \left ( {\partial S\over \partial n} \right )_{V,U} 

<math>~ \left ( {\partial T\over \partial S} \right )_V 

<math>~ \left ( {\partial T\over \partial S} \right )_P 

<math>~ -\left ( {\partial P\over \partial V} \right )_T 

[edit] Equation Table for an Ideal Gas (<math>PV^m=constant</math>)

[edit] other useful identities

<math>\Delta U = q_{by} + w_{on} = q_{by} - \int P_{ext}dV = q_{by} - P_{ext}\Delta V</math>

<math>H = U + PV \,\!</math>

<math>A = U - TS \,\!</math>

<math>G = H - TS = \sum_{i} \mu_{i} n_{i} \,\!</math>

<math>dU\left(S,V,{n_{i}}\right) = TdS - PdV + \sum_{i} \mu_{i} dn_i</math>

<math>dH\left(S,P,n_{i}\right) = TdS + VdP + \sum_{i} \mu_{i} dn_{i}</math>

<math>dA\left(T,V,n_{i}\right) = -SdT - PdV + \sum_{i} \mu_{i} dn_{i}</math>

<math>dG\left(T,P,n_{i}\right) = -SdT + VdP + \sum_{i} \mu_{i} dn_{i}</math>

<math>C_V = \left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V</math>

<math>C_P = \left(\frac{\partial H}{\partial T}\right)_P</math>

<math>\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H</math>

<math>\kappa_{T} = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T</math>

<math>\alpha_{P} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\left(\frac{\partial H}{\partial P}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_P</math>

<math>\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P</math>

<math>H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_P</math>

<math>U = -T^2\left(\frac{\partial \left(A/T\right)}{\partial T}\right)_V</math>

[edit] Proof #1

An example using the above methods is:

<math>

\left(\frac{\partial T}{\partial P}\right)_H = -\frac{1}{C_P}

  \left(\frac{\partial H}{\partial P}\right)_T

</math>

<math>

\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 </math>

<math>

\left(\frac{\partial T}{\partial P}\right)_H = -\left(\frac{\partial H}{\partial P}\right)_T

  \left(\frac{\partial T}{\partial H}\right)_P

</math>

<math>

= \frac{-1}{\left(\frac{\partial H}{\partial T}\right)_P}

 \left(\frac{\partial H}{\partial P}\right)_T

</math> ; <math>C_P = \left(\frac{\partial H}{\partial T}\right)_P</math>

<math>

\Rightarrow \left(\frac{\partial T}{\partial P}\right)_H = -\frac{1}{C_P}

  \left(\frac{\partial H}{\partial P}\right)_T

</math>

[edit] Proof #2

Another example:

<math>

C_V = T\left(\frac{\partial S}{\partial T}\right)_V </math>

<math>

U = q + w \,\! </math>

<math>

dU = dq_{rev} + w_{rev} ; dS = \frac{dq_{rev}}{T}, w_{rev} = -PdV \,\! </math>

<math>

= TdS-PdV \,\! </math>

<math>

\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 </math>

<math>

\Rightarrow C_V = T\left(\frac{\partial S}{\partial T}\right)_V </math>

[edit] References

• 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.
• 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.