Thermodynamic integration is used to calculate the difference in the Helmholtz energy function,
, between two states.
The path must be continuous and reversible (Ref. 1 Eq. 3.5)
![{\displaystyle \Delta A=A(\lambda )-A(\lambda _{0})=\int _{\lambda _{0}}^{\lambda }\left\langle {\frac {\partial U(\mathbf {r} ,\lambda )}{\partial \lambda }}\right\rangle _{\lambda }~\mathrm {d} \lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/708863ad5d4728db0f7401e10a84346af9f67d18)
Isothermal integration
At constant temperature (Ref. 2 Eq. 5):
![{\displaystyle {\frac {A(\rho _{2},T)}{Nk_{B}T}}={\frac {A(\rho _{1},T)}{Nk_{B}T}}+\int _{\rho _{1}}^{\rho _{2}}{\frac {p(\rho )}{k_{B}T\rho ^{2}}}~\mathrm {d} \rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8164571aa9e2e4e6bcc738260067c9b3298f05d2)
Isobaric integration
At constant pressure (Ref. 2 Eq. 6):
![{\displaystyle {\frac {G(T_{2},p)}{Nk_{B}T_{2}}}={\frac {G(T_{1},p)}{Nk_{B}T_{1}}}-\int _{T_{1}}^{T_{2}}{\frac {H(T)}{Nk_{B}T^{2}}}~\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed412d53295c8f698d605d7df6afe8dd5d31c3e1)
where
is the Gibbs energy function and
is the enthalpy.
Isochoric integration
At constant volume (Ref. 2 Eq. 7):
![{\displaystyle {\frac {A(T_{2},V)}{Nk_{B}T_{2}}}={\frac {A(T_{1},V)}{Nk_{B}T_{1}}}-\int _{T_{1}}^{T_{2}}{\frac {U(T)}{Nk_{B}T^{2}}}~\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/807c53f6c18f5290cb7cc87f91744e09e993dbad)
where
is the internal energy.
See also
References
- J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics 48 pp. 587 - 671 (1976)
- C. Vega, E. Sanz, J. L. F. Abascal and E. G. Noya "Determination of phase diagrams via computer simulation: methodology and applications to water, electrolytes and proteins", Journal of Physics: Condensed Matter 20 153101 (2008) (section 4)