# thermo.lennard_jones module¶

thermo.lennard_jones.Stockmayer(Tm=None, Tb=None, Tc=None, Zc=None, omega=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s Stockmayer parameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 7 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, omega (1966) for chemicals which don’t.

Parameters: Returns: Tm : float, optional Melting temperature of fluid [K] Tb : float, optional Boiling temperature of fluid [K] Tc : float, optional Critical temperature, [K] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] CASRN : string, optional CASRN [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Method : string, optional A string for the method name to use, as defined by constants in Stockmayer_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain epsilon for the desired chemical, and will return methods instead of epsilon

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [R423], for 322 chemicals.

References

 [R422] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006
 [R423] (1, 2) Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

Examples

>>> Stockmayer(CASRN='64-17-5')
1291.41

thermo.lennard_jones.molecular_diameter(Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s L-J molecular diameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 9 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966) for chemicals which don’t.

Parameters: Returns: Tc : float, optional Critical temperature, [K] Pc : float, optional Critical pressure, [Pa] Vc : float, optional Critical volume, [m^3/mol] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] Vm : float, optional Molar volume of liquid at the melting point of the fluid [K] Vb : float, optional Molar volume of liquid at the boiling point of the fluid [K] CASRN : string, optional CASRN [-] sigma : float Lennard-Jones molecular diameter, [Angstrom] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Method : string, optional A string for the method name to use, as defined by constants in molecular_diameter_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain sigma for the desired chemical, and will return methods instead of sigma

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [R425], for 322 chemicals.

References

 [R424] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006
 [R425] (1, 2) Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

Examples

>>> molecular_diameter(CASRN='64-17-5')
4.23738

thermo.lennard_jones.sigma_Flynn(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [R426] as reported in [R427].

$\sigma = 0.561(V_c^{1/3})^{5/4}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

 [R426] (1, 2) Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
 [R427] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

>>> sigma_Flynn(0.000268)
5.2506948422196285

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R428].

$\sigma = 2.44(T_c/P_c)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of critical pressure are atmospheres.

References

 [R428] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_2(560.1, 4550000)
5.658657684653222

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_1(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [R429].

$\sigma = 0.841 V_c^{1/3}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vc are mL/mol.

References

 [R429] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_1(0.000268)
5.422184116631474

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_boiling(Vb)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of liquid at boiling. CSP method by [R430].

$\sigma = 1.166V_{b,liq}^{1/3}$
Parameters: Vb : float Boiling molar volume of liquid [m^3/mol] sigma : float Lennard-Jones collision integral, [Angstrom]

Notes

Original units of Vb are mL/mol.

References

 [R430] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_boiling(0.0001015)
5.439018856944655

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_melting(Vm)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of a liquid at its melting point. CSP method by [R431].

$\sigma = 1.222 V_{m,sol}^{1/3}$
Parameters: Vm : float Melting molar volume of a liquid at its melting point [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vm are mL/mol.

References

 [R431] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_melting(8.8e-05)
5.435407341351406

thermo.lennard_jones.sigma_Stiel_Thodos(Vc, Zc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume and compressibility. CSP method by [R432].

$\sigma = 0.1866 V_c^{1/3} Z_c^{-6/5}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] Zc : float Critical compressibility of fluid, [-] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

 [R432] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

Monofluorobenzene

>>> sigma_Stiel_Thodos(0.000271, 0.265)
5.94300853971033

thermo.lennard_jones.sigma_Tee_Gotoh_Steward_1(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R433].

$\sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [R433] but are not included here, except for sigma_Tee_Gotoh_Steward_2.

References

 [R433] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> sigma_Tee_Gotoh_Steward_1(560.1, 4550000)
5.48402779790962

thermo.lennard_jones.sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature, pressure, and acentric factor. CSP method by [R434].

$\sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] omega : float Acentric factor for fluid, [-] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [R434] but are not included here, except for sigma_Tee_Gotoh_Steward_1.

References

 [R434] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> sigma_Tee_Gotoh_Steward_2(560.1, 4550000, 0.245)
5.412104867264477

thermo.lennard_jones.sigma_Silva_Liu_Macedo(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R435].

$\sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right) - 0.049029\left( \frac{T_c}{P_c}\right)^2$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Pc is originally in bar. An excellent paper. None is returned if the polynomial returns a negative number, as in the case of 1029.13 K and 3.83 bar.

References

 [R435] (1, 2) Silva, Carlos M., Hongqin Liu, and Eugenia A. Macedo. “Models for Self-Diffusion Coefficients of Dense Fluids, Including Hydrogen-Bonding Substances.” Chemical Engineering Science 53, no. 13 (July 1, 1998): 2423-29. doi:10.1016/S0009-2509(98)00037-2

Examples

>>> sigma_Silva_Liu_Macedo(560.1, 4550000)
5.164483998730177

thermo.lennard_jones.epsilon_Flynn(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [R436] as reported in [R437].

$\epsilon/k = 1.77 T_c^{5/6}$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R436] (1, 2) Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
 [R437] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

>>> epsilon_Flynn(560.1)
345.2984087011443

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_critical(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [R438].

$\epsilon/k = 0.77T_c$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R438] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_critical(560.1)
431.27700000000004

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_boiling(Tb)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses boiling temperature. CSP method by [R439].

$\epsilon/k = 1.15 T_b$
Parameters: Tb : float Boiling temperature [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R439] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_boiling(357.85)
411.5275

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_melting(Tm)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses melting temperature. CSP method by [R440].

$\epsilon/k = 1.92T_m$
Parameters: Tm : float Melting temperature [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R440] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_melting(231.15)
443.808

thermo.lennard_jones.epsilon_Stiel_Thodos(Tc, Zc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature and critical compressibility. CSP method by [R441].

$\epsilon/k = 65.3 T_c Z_c^{3.6}$
Parameters: Tc : float Critical temperature of fluid [K] Zc : float Critical compressibility of fluid, [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R441] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

Fluorobenzene

>>> epsilon_Stiel_Thodos(358.5, 0.265)
196.3755830305783

thermo.lennard_jones.epsilon_Tee_Gotoh_Steward_1(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature. CSP method by [R442].

$\epsilon/k = 0.7740T_c$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [R442] but are not included here, except for epsilon_Tee_Gotoh_Steward_2.

References

 [R442] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> epsilon_Tee_Gotoh_Steward_1(560.1)
433.5174

thermo.lennard_jones.epsilon_Tee_Gotoh_Steward_2(Tc, omega)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature and acentric factor. CSP method by [R443].

$\epsilon/k = (0.7915 + 0.1693 \omega)T_c$
Parameters: Tc : float Critical temperature of fluid [K] omega : float Acentric factor for fluid, [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [R443] but are not included here, except for epsilon_Tee_Gotoh_Steward_1.

References

 [R443] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> epsilon_Tee_Gotoh_Steward_2(560.1, 0.245)
466.55125785

thermo.lennard_jones.collision_integral_Neufeld_Janzen_Aziz(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 100. Values are accurate to 0.1 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [R444].

$\Omega_D = \frac{A}{T^{*B}} + \frac{C}{\exp(DT^*)} + \frac{E}{\exp(FT^{*})} + \frac{G}{\exp(HT^*)} + RT^{*B}\sin(ST^{*W}-P)$
Parameters: Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Omega : float Collision integral of A and B

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

$T^* = \frac{k_b T}{\epsilon}$

Results are very similar to those of the more modern formulation, collision_integral_Kim_Monroe.

Calculations begin to yield overflow errors in some values of (l, 2) after Tstar = 75, beginning with (1, 7). Also susceptible are (1, 5) and (1, 6).

References

 [R444] (1, 2) Neufeld, Philip D., A. R. Janzen, and R. A. Aziz. “Empirical Equations to Calculate 16 of the Transport Collision Integrals Omega(l, S)* for the Lennard-Jones (12-6) Potential.” The Journal of Chemical Physics 57, no. 3 (August 1, 1972): 1100-1102. doi:10.1063/1.1678363

Examples

>>> collision_integral_Neufeld_Janzen_Aziz(100, 1, 1)
0.516717697672334

thermo.lennard_jones.collision_integral_Kim_Monroe(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 400. Values are accurate to 0.007 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [R446].

$\Omega^{(l,s)*} = A^{(l,s)} + \sum_{k=1}^6 \left[ \frac{B_k^{(l,s)}} {(T^*)^k} + C_k^{(l,s)} (\ln T^*)^k \right]$
Parameters: Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Omega : float Collision integral of A and B

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

$T^* = \frac{k_b T}{\epsilon}$

References

 [R446] (1, 2) Kim, Sun Ung, and Charles W. Monroe. “High-Accuracy Calculations of Sixteen Collision Integrals for Lennard-Jones (12-6) Gases and Their Interpolation to Parameterize Neon, Argon, and Krypton.” Journal of Computational Physics 273 (September 15, 2014): 358-73. doi:10.1016/j.jcp.2014.05.018.

Examples

>>> collision_integral_Kim_Monroe(400, 1, 1)
0.4141818082392228

thermo.lennard_jones.Tstar(T, epsilon_k=None, epsilon=None)[source]

This function calculates the parameter Tstar as needed in performing collision integral calculations.

$T^* = \frac{kT}{\epsilon}$
Parameters: epsilon_k : float, optional Lennard-Jones depth of potential-energy minimum over k, [K] epsilon : float, optional Lennard-Jones depth of potential-energy minimum [J] Tstar : float Dimentionless temperature for calculating collision integral, [-]

Notes

Tabulated values are normally listed as epsilon/k. k is the Boltzman constant, with units of J/K.

References

 [R448] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> Tstar(T=318.2, epsilon_k=308.43)
1.0316765554582887