Thermodynamic integration
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)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 }
Isothermal integration
At constant temperature (Ref. 2 Eq. 5):
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 }
Isobaric integration
At constant pressure (Ref. 2 Eq. 6):
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}
where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G} is the Gibbs energy function and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is the enthalpy.
Isochoric integration
At constant volume (Ref. 2 Eq. 7):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A(T_2,V)}{Nk_BT_2} = \frac{A(T_1,V)}{Nk_BT_1} - \int_{T_1}^{T_2} \frac{U(T)}{Nk_BT^2} ~\mathrm{d}T }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} 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)