Thermodynamics Equations

OTHER USEFUL IDENTITIES

:

\Delta U = q_{by} + w_{on} = q_{by} - \int P_{ext}dV = q_{by} - 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}}
ight) = TdS - PdV + \sum_{i} \mu_{i} dn_i

dH\left(S,P,n_{i}
ight) = TdS + VdP + \sum_{i} \mu_{i} dn_{i}

dA\left(T,V,n_{i}
ight) = -SdT - PdV + \sum_{i} \mu_{i} dn_{i}

dG\left(T,P,n_{i}
ight) = -SdT + VdP + \sum_{i} \mu_{i} dn_{i}

C_v = \left(rac{\partial U}{\partial T}
ight)_V = T\left(rac{\partial S}{\partial T}
ight)_V

C_p = \left(rac{\partial H}{\partial T}
ight)_P

\mu_{JT} = \left(rac{\partial T}{\partial P}
ight)_H

\kappa_{T} = -rac{1}{V}\left(rac{\partial V}{\partial P}
ight)_T

\alpha_{P} = rac{1}{V}\left(rac{\partial V}{\partial T}
ight)_P

\left(rac{\partial H}{\partial P}
ight)_T = V - T\left(rac{\partial V}{\partial T}
ight)_P

\left(rac{\partial U}{\partial V}
ight)_T = T\left(rac{\partial P}{\partial T}
ight)_V - P

H = -T^2\left(rac{\partial \left(G/T
ight)}{\partial T}
ight)_P

U = -T^2\left(rac{\partial \left(A/T
ight)}{\partial T}
ight)_V

Proof #1

An example using the above methods is:

:

\left(rac{\partial T}{\partial P}
ight)_H 
= -rac{1}{C_p}
   \left(rac{\partial H}{\partial P}
ight)_T

\left(rac{\partial T}{\partial P}
ight)_H
\left(rac{\partial P}{\partial H}
ight)_T
\left(rac{\partial H}{\partial T}
ight)_P
= -1

\left(rac{\partial T}{\partial P}
ight)_H 
= -\left(rac{\partial H}{\partial P}
ight)_T
   \left(rac{\partial T}{\partial H}
ight)_P

::::

= rac{-1}{\left(rac{\partial H}{\partial T}
ight)_P}
  \left(rac{\partial H}{\partial P}
ight)_T

;

C_p = \left(rac{\partial H}{\partial T}
ight)_P

\Rightarrow \left(rac{\partial T}{\partial P}
ight)_H
= -rac{1}{C_p}
   \left(rac{\partial H}{\partial P}
ight)_T

Proof #2

Another example:

:

C_v = T\left(rac{\partial S}{\partial T}
ight)_V

U = q + w \,\!

dU = dq_{rev} + w_{rev} ; dS = rac{dq_{rev}}{T}, w_{rev} = -PdV \,\!

= TdS-PdV  \,\!

\left(rac{\partial U}{\partial T}
ight)_V
= T\left(rac{\partial S}{\partial T}
ight)_V
- P\left(rac{\partial V}{\partial T}
ight)_V ; C_v = \left(rac{\partial U}{\partial T}
ight)_V

\Rightarrow C_v = T\left(rac{\partial S}{\partial T}
ight)_V

REFERENCES

Peter Atkins and Julio de Paula, ''Physical Chemistry'', 7th edition, W.H. Freeman and Company, 2002 0-7167-3539-3 .

Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.

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