Thermodynamic integration: Difference between revisions
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'''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]], <math>A</math>, between two states. | '''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]], <math>A</math>, between two states. | ||
The path must be ''continuous'' and ''reversible''. | The path '''must''' be ''continuous'' and ''reversible'' (Ref. 1 Eq. 3.5) | ||
:<math>\Delta A = A - | :<math>\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</math> | ||
==Isothermal integration== | ==Isothermal integration== | ||
Ref. | At constant [[temperature]] (Ref. 2 Eq. 5): | ||
:<math>\frac{A(\rho_2,T)}{Nk_BT} = \frac{A(\rho_1,T)}{Nk_BT} + \int_{\rho_1}^{\rho_2} \frac{p(\rho)}{k_B T \rho^2} ~\mathrm{d}\rho </math> | :<math>\frac{A(\rho_2,T)}{Nk_BT} = \frac{A(\rho_1,T)}{Nk_BT} + \int_{\rho_1}^{\rho_2} \frac{p(\rho)}{k_B T \rho^2} ~\mathrm{d}\rho </math> | ||
==Isobaric integration== | ==Isobaric integration== | ||
Ref. | At constant [[pressure]] (Ref. 2 Eq. 6): | ||
:<math>\frac{G(T_2,p)}{Nk_BT_2} = \frac{G(T_1,p)}{Nk_BT_1} - \int_{T_1}^{T_2} \frac{H(T)}{Nk_BT^2} ~\mathrm{d}T </math> | :<math>\frac{G(T_2,p)}{Nk_BT_2} = \frac{G(T_1,p)}{Nk_BT_1} - \int_{T_1}^{T_2} \frac{H(T)}{Nk_BT^2} ~\mathrm{d}T </math> | ||
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where <math>G</math> is the [[Gibbs energy function]] and <math>H</math> is the [[enthalpy]]. | where <math>G</math> is the [[Gibbs energy function]] and <math>H</math> is the [[enthalpy]]. | ||
==Isochoric integration== | ==Isochoric integration== | ||
Ref. | At constant volume (Ref. 2 Eq. 7): | ||
:<math>\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 </math> | :<math>\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 </math> | ||
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*[[Gibbs-Duhem integration]] | *[[Gibbs-Duhem integration]] | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1103/RevModPhys.48.587 J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics '''48''' pp. 587 - 671 (1976)] | |||
#[http://dx.doi.org/10.1088/0953-8984/20/15/153101 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) | #[http://dx.doi.org/10.1088/0953-8984/20/15/153101 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) | ||
[[category:classical thermodynamics]] | [[category:classical thermodynamics]] | ||
Revision as of 11:44, 5 August 2008
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 (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 \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 (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(\rho_2,T)}{Nk_BT} = \frac{A(\rho_1,T)}{Nk_BT} + \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 (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{G(T_2,p)}{Nk_BT_2} = \frac{G(T_1,p)}{Nk_BT_1} - \int_{T_1}^{T_2} \frac{H(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 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)