Lennard-Jones Models (chemicals.lennard_jones)

This module contains lookup functions and estimation methods for the parameters molecular diameter sigma and the Stockmayer parameter epsilon. These are used for diffusivity calculations. It also contains several methods for computing the collision integral, another parameter used in the Lennard-Jones model.

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Stockmayer Parameter

chemicals.lennard_jones.Stockmayer(CASRN='', Tm=None, Tb=None, Tc=None, Zc=None, omega=None, method=None)

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.

Preferred 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

CASRNstr, optional

CASRN [-]

Tmfloat, optional

Melting temperature of compound [K]

Tbfloat, optional

Boiling temperature of compound [K]

Tcfloat, optional

Critical temperature of compound, [K]

Zcfloat, optional

Critical compressibility of compound, [-]

omegafloat, optional

Acentric factor of compound, [-]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

Other Parameters

methodstr, optional

A string for the method name to use, as defined by constants in Stockmayer_all_methods

Notes

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

The tabulated data is from [2], for 322 chemicals.

References

1

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

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
>>> Stockmayer('7727-37-9')
71.4

chemicals.lennard_jones.Stockmayer_methods(CASRN=None, Tm=None, Tb=None, Tc=None, Zc=None, omega=None)

Return all methods available to obtain the Stockmayer parameter for the desired chemical.

Parameters

CASRNstr, optional

CASRN [-]

Tmfloat, optional

Melting temperature of compound [K]

Tbfloat, optional

Boiling temperature of compound [K]

Tcfloat, optional

Critical temperature of compound, [K]

Zcfloat, optional

Critical compressibility of compound, [-]

omegafloat, optional

Acentric factor of compound, [-]

Returns

methodslist[str]

Methods which can be used to obtain Stockmayer with the given inputs.

See also

Stockmayer

chemicals.lennard_jones.Stockmayer_all_methods = ('Magalhães, Lito, Da Silva, and Silva (2013)', 'Poling et al. (2001)', 'Tee, Gotoh, and Stewart CSP with Tc, omega (1966)', 'Stiel and Thodos Tc, Zc (1962)', 'Flynn (1960)', 'Bird, Stewart, and Light (2002) critical relation', 'Tee, Gotoh, and Stewart CSP with Tc (1966)', 'Bird, Stewart, and Light (2002) boiling relation', 'Bird, Stewart, and Light (2002) melting relation')

Tuple of method name keys. See the Stockmayer for the actual references

Stockmayer Parameter Correlations

chemicals.lennard_jones.epsilon_Flynn(Tc)

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1] as reported in [2].

$$\epsilon/k = 1.77 T_c^{5/6}$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

References

1

Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).

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

chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_critical(Tc)

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1].

$$\epsilon/k = 0.77T_c$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

References

1

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

chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_boiling(Tb)

Calculates Lennard-Jones depth of potential-energy minimum. Uses boiling temperature. CSP method by [1].

$$\epsilon/k = 1.15 T_b$$

Parameters

Tbfloat

Boiling temperature [K]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

References

1

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

chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_melting(Tm)

Calculates Lennard-Jones depth of potential-energy minimum. Uses melting temperature. CSP method by [1].

$$\epsilon/k = 1.92T_m$$

Parameters

Tmfloat

Melting temperature [K]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

References

1

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

chemicals.lennard_jones.epsilon_Stiel_Thodos(Tc, Zc)

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature and critical compressibility. CSP method by [1].

$$\epsilon/k = 65.3 T_c Z_c^{3.6}$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Zcfloat

Critical compressibility of fluid, [-]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

References

1

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

chemicals.lennard_jones.epsilon_Tee_Gotoh_Steward_1(Tc)

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature. CSP method by [1].

$$\epsilon/k = 0.7740T_c$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [1] but are not included here, except for epsilon_Tee_Gotoh_Steward_2.

References

1(1,2)

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

chemicals.lennard_jones.epsilon_Tee_Gotoh_Steward_2(Tc, omega)

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature and acentric factor. CSP method by [1].

$$\epsilon/k = (0.7915 + 0.1693 \omega)T_c$$

Parameters

Tcfloat

Critical temperature of fluid [K]

omegafloat

Acentric factor for fluid, [-]

Returns

epsilon_kfloat

Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [1] but are not included here, except for epsilon_Tee_Gotoh_Steward_1.

References

1(1,2)

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

Molecular Diameter

chemicals.lennard_jones.molecular_diameter(CASRN=None, Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None, method=None)

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.

Preferred 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

CASRNstr, optional

CASRN [-]

Tcfloat, optional

Critical temperature, [K]

Pcfloat, optional

Critical pressure, [Pa]

Vcfloat, optional

Critical volume, [m^3/mol]

Zcfloat, optional

Critical compressibility, [-]

omegafloat, optional

Acentric factor of compound, [-]

Vmfloat, optional

Molar volume of liquid at the melting point of the fluid [K]

Vbfloat, optional

Molar volume of liquid at the boiling point of the fluid [K]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Other Parameters

methodstr, optional

A string for the method name to use, as defined by constants in molecular_diameter_all_methods

Notes

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

The tabulated data is from [2], for 322 chemicals.

References

1

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

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
>>> molecular_diameter('7727-37-9')
3.798

chemicals.lennard_jones.molecular_diameter_methods(CASRN=None, Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None)

Return all methods available to obtain the molecular diameter for the desired chemical.

Parameters

CASRNstr, optional

CASRN [-]

Tcfloat, optional

Critical temperature, [K]

Pcfloat, optional

Critical pressure, [Pa]

Vcfloat, optional

Critical volume, [m^3/mol]

Zcfloat, optional

Critical compressibility, [-]

omegafloat, optional

Acentric factor of compound, [-]

Vmfloat, optional

Molar volume of liquid at the melting point of the fluid [K]

Vbfloat, optional

Molar volume of liquid at the boiling point of the fluid [K]

Returns

methodslist[str]

Methods which can be used to obtain molecular_diameter with the given inputs.

See also

molecular_diameter

chemicals.lennard_jones.molecular_diameter_all_methods = ('Magalhães, Lito, Da Silva, and Silva (2013)', 'Poling et al. (2001)', 'Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966)', 'Silva, Liu, and Macedo (1998) critical relation with Tc, Pc', 'Bird, Stewart, and Light (2002) critical relation with Tc, Pc', 'Tee, Gotoh, and Stewart CSP with Tc, Pc (1966)', 'Stiel and Thodos Vc, Zc (1962)', 'Flynn (1960)', 'Bird, Stewart, and Light (2002) critical relation with Vc', 'Bird, Stewart, and Light (2002) boiling relation', 'Bird, Stewart, and Light (2002) melting relation')

Tuple of method name keys. See the molecular_diameter for the actual references

Molecular Diameter Correlations

chemicals.lennard_jones.sigma_Flynn(Vc)

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1] as reported in [2].

$$\sigma = 0.561(V_c^{1/3})^{5/4}$$

Parameters

Vcfloat

Critical volume of fluid [m^3/mol]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

1

Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).

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

chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc)

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1].

$$\sigma = 2.44(T_c/P_c)^{1/3}$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of critical pressure are atmospheres.

References

1

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

chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_1(Vc)

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1].

$$\sigma = 0.841 V_c^{1/3}$$

Parameters

Vcfloat

Critical volume of fluid [m^3/mol]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vc are mL/mol.

References

1

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

chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_boiling(Vb)

Calculates Lennard-Jones molecular diameter. Uses molar volume of liquid at boiling. CSP method by [1].

$$\sigma = 1.166V_{b,liq}^{1/3}$$

Parameters

Vbfloat

Boiling molar volume of liquid [m^3/mol]

Returns

sigmafloat

Lennard-Jones collision integral, [Angstrom]

Notes

Original units of Vb are mL/mol.

References

1

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

chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_melting(Vm)

Calculates Lennard-Jones molecular diameter. Uses molar volume of a liquid at its melting point. CSP method by [1].

$$\sigma = 1.222 V_{m,sol}^{1/3}$$

Parameters

Vmfloat

Melting molar volume of a liquid at its melting point [m^3/mol]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vm are mL/mol.

References

1

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

chemicals.lennard_jones.sigma_Stiel_Thodos(Vc, Zc)

Calculates Lennard-Jones molecular diameter. Uses critical volume and compressibility. CSP method by [1].

$$\sigma = 0.1866 V_c^{1/3} Z_c^{-6/5}$$

Parameters

Vcfloat

Critical volume of fluid [m^3/mol]

Zcfloat

Critical compressibility of fluid, [-]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

1

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

chemicals.lennard_jones.sigma_Tee_Gotoh_Steward_1(Tc, Pc)

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1].

$$\sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3}$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [1] but are not included here, except for sigma_Tee_Gotoh_Steward_2.

References

1(1,2)

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

chemicals.lennard_jones.sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)

Calculates Lennard-Jones molecular diameter. Uses critical temperature, pressure, and acentric factor. CSP method by [1].

$$\sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3}$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

omegafloat

Acentric factor for fluid, [-]

Returns

sigmafloat

Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [1] but are not included here, except for sigma_Tee_Gotoh_Steward_1.

References

1(1,2)

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

chemicals.lennard_jones.sigma_Silva_Liu_Macedo(Tc, Pc)

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1].

$$\sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right) - 0.049029\left( \frac{T_c}{P_c}\right)^2$$

Parameters

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

Returns

sigmafloat

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

1

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

Utility Functions

chemicals.lennard_jones.T_star(T, epsilon_k=None, epsilon=None)

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

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

Parameters

epsilon_kfloat, optional

Lennard-Jones depth of potential-energy minimum over k, [K]

epsilonfloat, optional

Lennard-Jones depth of potential-energy minimum [J]

Returns

T_starfloat

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

1

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

Examples

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