Heat Capacity (chemicals.heat_capacity)

This module contains many heat capacity model equations, heat capacity estimation equations, enthalpy and entropy integrals of those heat capacity equations, enthalpy/entropy flash initialization routines, and many dataframes of coefficients.

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Gas Heat Capacity Model Equations

chemicals.heat_capacity.TRCCp(T, a0, a1, a2, a3, a4, a5, a6, a7)

Calculates ideal gas heat capacity using the model developed in [1]. The ideal gas heat capacity is given by:

$$C_p = R\left(a_0 + (a_1/T^2) \exp(-a_2/T) + a_3 y^2 + (a_4 - a_5/(T-a_7)^2 )y^j \right)$$

$$y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0$$

Parameters

Tfloat

Temperature [K]

a1-a7float

Coefficients

Returns

Cpfloat

Ideal gas heat capacity , [J/mol/K]

Notes

j is set to 8. Analytical integrals are available for this expression.

References

1

Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.

Examples

>>> TRCCp(300, 4.0, 7.65E5, 720., 3.565, -0.052, -1.55E6, 52., 201.)
42.065271080974654

chemicals.heat_capacity.TRCCp_integral(T, a0, a1, a2, a3, a4, a5, a6, a7, I=0)

Integrates ideal gas heat capacity using the model developed in [1]. Best used as a delta only. The difference in enthalpy with respect to 0 K is given by:

$$\frac{H(T) - H^{ref}}{RT} = a_0 + a_1x(a_2)/(a_2T) + I/T + h(T)/T$$

$$h(T) = (a_5 + a_7)\left[(2a_3 + 8a_4)\ln(1-y)+ \left\{a_3\left(1 + \frac{1}{1-y}\right) + a_4\left(7 + \frac{1}{1-y}\right)\right\}y + a_4\left\{3y^2 + (5/3)y^3 + y^4 + (3/5)y^5 + (1/3)y^6\right\} + (1/7)\left\{a_4 - \frac{a_5}{(a_6+a_7)^2}\right\}y^7\right]$$

$$h(T) = 0 \text{ for } T \le a_7 y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0$$

Parameters

Tfloat

Temperature [K]

a1-a7float

Coefficients

Ifloat, optional

Integral offset

Returns

H-H(0)float

Difference in enthalpy from 0 K , [J/mol]

Notes

Analytical integral as provided in [1] and verified with numerical integration.

References

1(1,2)

Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.

Examples

>>> TRCCp_integral(298.15, 4.0, 7.65E5, 720., 3.565, -0.052, -1.55E6, 52.,
... 201., 1.2)
10802.536262068483

chemicals.heat_capacity.TRCCp_integral_over_T(T, a0, a1, a2, a3, a4, a5, a6, a7, J=0)

Integrates ideal gas heat capacity over T using the model developed in [1]. Best used as a delta only. The difference in ideal-gas entropy with respect to 0 K is given by:

$$\frac{S^\circ}{R} = J + a_0\ln T + \frac{a_1}{a_2^2}\left(1 + \frac{a_2}{T}\right)x(a_2) + s(T) s(T) = \left[\left\{a_3 + \left(\frac{a_4 a_7^2 - a_5}{a_6^2}\right) \left(\frac{a_7}{a_6}\right)^4\right\}\left(\frac{a_7}{a_6}\right)^2 \ln z + (a_3 + a_4)\ln\left(\frac{T+a_6}{a_6+a_7}\right) +\sum_{i=1}^7 \left\{\left(\frac{a_4 a_7^2 - a_5}{a_6^2}\right)\left( \frac{-a_7}{a_6}\right)^{6-i} - a_4\right\}\frac{y^i}{i} - \left\{\frac{a_3}{a_6}(a_6 + a_7) + \frac{a_5 y^6}{7a_7(a_6+a_7)} \right\}y\right]$$

$$s(T) = 0 \text{ for } T \le a_7$$

$$z = \frac{T}{T+a_6} \cdot \frac{a_7 + a_6}{a_7}$$

$$y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0$$

Parameters

Tfloat

Temperature [K]

a1-a7float

Coefficients

Jfloat, optional

Integral offset

Returns

S-S(0)float

Difference in entropy from 0 K , [J/mol/K]

Notes

Analytical integral as provided in [1] and verified with numerical integration.

References

1(1,2)

Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.

Examples

>>> TRCCp_integral_over_T(300, 4.0, 124000, 245, 50.539, -49.469,
... 220440000, 560, 78)
213.80156219151888

chemicals.heat_capacity.Shomate(T, A, B, C, D, E)

Calculates heat capacity using the Shomate polynomial model [1]. The heat capacity is given by:

$$C_p = A + BT + CT^2 + DT^3 + \frac{E}{T^2}$$

Parameters

Tfloat

Temperature [K]

Afloat

Parameter, [J/(mol*K)]

Bfloat

Parameter, [J/(mol*K^2)]

Cfloat

Parameter, [J/(mol*K^3)]

Dfloat

Parameter, [J/(mol*K^4)]

Efloat

Parameter, [J*K/(mol)]

Returns

Cpfloat

Heat capacity , [J/mol/K]

Notes

Analytical integrals are available for this expression. In some sources such as [1], the equation is written with temperature in units of kilokelvin. The coefficients can be easily adjusted to be in the proper SI form.

References

1(1,2,3)

Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds., NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q

Examples

Coefficients for water vapor from [1]:

>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate(500, *water_low_gas_coeffs)
35.21836175

chemicals.heat_capacity.Shomate_integral(T, A, B, C, D, E)

Calculates the enthalpy integral using the Shomate polynomial model [1]. The difference in enthalpy with respect to 0 K is given by:

$${H(T) - H^{0}} = A T + \frac{B T^{2}}{2} + \frac{C T^{3}}{3} + \frac{D T^{4}}{4} - \frac{E}{T}$$

Parameters

Tfloat

Temperature [K]

Afloat

Parameter, [J/(mol*K)]

Bfloat

Parameter, [J/(mol*K^2)]

Cfloat

Parameter, [J/(mol*K^3)]

Dfloat

Parameter, [J/(mol*K^4)]

Efloat

Parameter, [J*K/(mol)]

Returns

H-H(0)float

Difference in enthalpy from 0 K , [J/mol]

References

1(1,2)

Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds., NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q

Examples

Coefficients for water vapor from [1]:

>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate_integral(500, *water_low_gas_coeffs)
15979.2447

chemicals.heat_capacity.Shomate_integral_over_T(T, A, B, C, D, E)

Integrates the heat capacity over T using the model developed in [1]. The difference in entropy with respect to 0 K is given by:

$$s(T) = A \log{\left(T \right)} + B T + \frac{C T^{2}}{2} + \frac{D T^{3}}{3} - \frac{E}{2 T^{2}}$$

Parameters

Tfloat

Temperature [K]

Afloat

Parameter, [J/(mol*K)]

Bfloat

Parameter, [J/(mol*K^2)]

Cfloat

Parameter, [J/(mol*K^3)]

Dfloat

Parameter, [J/(mol*K^4)]

Efloat

Parameter, [J*K/(mol)]

Returns

S-S(0)float

Difference in entropy from 0 K , [J/mol/K]

References

1(1,2)

Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds., NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q

Examples

Coefficients for water vapor from [1]:

>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate_integral_over_T(500, *water_low_gas_coeffs)
191.00554

class chemicals.heat_capacity.ShomateRange(coeffs, Tmin, Tmax)

Implementation of a range of the Shomate equation presented in [1] for calculating the heat capacity of a chemical. Implements the enthalpy and entropy integrals as well.

Parameters

coeffslist[float]

Six coefficients for the equation, [-]

Tminfloat

Minimum temperature any experimental data was available at, [K]

Tmaxfloat

Maximum temperature any experimental data was available at, [K]

References

1

Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds., NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q

Methods

calculate(T)	Return heat capacity as a function of temperature.
calculate_integral(Ta, Tb)	Return the enthalpy integral of heat capacity from Ta to Tb.
calculate_integral_over_T(Ta, Tb)	Return the entropy integral of heat capacity from Ta to Tb.

calculate(T)

Return heat capacity as a function of temperature.

Parameters

Tfloat

Temperature, [K]

Returns

Cpfloat

Liquid heat capacity as T, [J/mol/K]

calculate_integral(Ta, Tb)

Return the enthalpy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dHfloat

Enthalpy difference between Ta and Tb, [J/mol]

calculate_integral_over_T(Ta, Tb)

Return the entropy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dSfloat

Entropy difference between Ta and Tb, [J/mol/K]

chemicals.heat_capacity.Poling(T, a, b, c, d, e)

Return the ideal-gas molar heat capacity of a chemical using polynomial regressed coefficients as described by Poling et. al. [1].

Parameters

Tfloat

Temperature, [K]

a,b,c,d,efloat

Regressed coefficients.

Returns

Cpgmfloat

Gas molar heat capacity, [J/mol/K]

See also

Poling_integral

Poling_integral_over_T

Notes

The ideal gas heat capacity is given by:

$$C_n = R*(a + bT + cT^2 + dT^3 + eT^4)$$

The data is based on the Poling data bank.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

Examples

Compute the gas heat capacity of Methane at 300 K:

>>> Poling(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
35.850973388425

chemicals.heat_capacity.Poling_integral(T, a, b, c, d, e)

Return the integral of the ideal-gas constant-pressure heat capacity of a chemical using polynomial regressed coefficients as described by Poling et. al. [1].

Parameters

Tfloat

Temperature, [K]

a,b,c,d,efloat

Regressed coefficients.

Returns

Hfloat

Difference in enthalpy from 0 K, [J/mol]

See also

Poling

Poling_integral_over_T

Notes

Integral was computed with SymPy.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

Examples

Compute the gas enthalpy of Methane at 300 K (with reference to 0 K):

>>> Poling_integral(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
10223.67533722261

chemicals.heat_capacity.Poling_integral_over_T(T, a, b, c, d, e)

Return the integral over temperature of the ideal-gas constant-pressure heat capacity of a chemical using polynomial regressed coefficients as described by Poling et. al. [1].

Parameters

Tfloat

Temperature, [K]

a,b,c,d,efloat

Regressed coefficients.

Returns

Sfloat

Difference in entropy from 0 K, [J/mol/K]

See also

Poling

Poling_integral

Notes

Integral was computed with SymPy.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

Examples

Compute the gas entropy of Methane at 300 K (with reference to 0 K):

>>> Poling_integral_over_T(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
205.46526328058

chemicals.heat_capacity.PPDS2(T, Ts, C_low, C_inf, a1, a2, a3, a4, a5)

Calculates the ideal-gas heat capacity using the [1] emperical (parameter-regressed) method, called the PPDS 2 equation for heat capacity.

\frac{C_p^0}{R} = C_{low} + (C_\inf - C_{low})y^2\left(1 + (y-1) \left[\sum_{i=0}^4 a_i y^i\right]\right)

$$y = \frac{T}{T + T_s}$$

Parameters

Tfloat

Temperature of fluid [K]

Tsfloat

Fit temperature; no physical meaning [K]

C_lowfloat

Fit parameter equal to Cp/R at a low temperature, [-]

C_inffloat

Fit parameter equal to Cp/R at a high temperature, [-]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

a3float

Regression parameter, [-]

a4float

Regression parameter, [-]

a5float

Regression parameter, [-]

Returns

Cpgmfloat

Gas molar heat capacity, [J/mol/K]

References

1(1,2)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/TDE_Help/Eqns-Pure-Cp0/PPDS2Cp0.htm.

Examples

n-pentane at 350 K from [1]

>>> PPDS2(T=350.0, Ts=462.493, C_low=4.54115, C_inf=9.96847, a1=-103.419, a2=695.484, a3=-2006.1, a4=2476.84, a5=-1186.47)
136.46338956689

Gas Heat Capacity Estimation Models

chemicals.heat_capacity.Lastovka_Shaw(T, similarity_variable, cyclic_aliphatic=False, MW=None, term_A=None)

Calculate ideal-gas constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

$$term_A = A1 + A2*a \text{ if cyclic aliphatic}$$

$$term_A = \left(A_2 + \frac{A_1 - A_2}{1 + \exp(\frac{\alpha-A_3}{A_4})}\right) \text{ if not cyclic aliphatic}$$

$$C_p^0 = term_A + (B_{11} + B_{12}\alpha)\left(-\frac{(C_{11} + C_{12}\alpha)}{T}\right)^2 \frac{\exp(-(C_{11} + C_{12}\alpha)/T)}{[1-\exp(-(C_{11}+C_{12}\alpha)/T)]^2} + (B_{21} + B_{22}\alpha)\left(-\frac{(C_{21} + C_{22}\alpha)}{T}\right)^2 \frac{\exp(-(C_{21} + C_{22}\alpha)/T)}{[1-\exp(-(C_{21}+C_{22}\alpha)/T)]^2}$$

Parameters

Tfloat

Temperature of gas [K]

similarity_variablefloat

Similarity variable as defined in [1], [mol/g]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

MWfloat, optional

Molecular weight, [g/mol]

term_Afloat, optional

Term A in Lastovka-Shaw equation, [J/g]

Returns

Cpgfloat

Gas constant-pressure heat capacity, J/mol/K if MW given; J/kg/K otherwise

Notes

Original model is in terms of J/g/K.

A1 = -0.1793547 text{ if cyclic aliphatic}

A1 = 0.58 text{ if not cyclic aliphatic}

A2 = 3.86944439 text{ if cyclic aliphatic}

A2 = 1.25 text{ if not cyclic aliphatic}

A3 = 0.17338003

A4 = 0.014

B11 = 0.73917383

B12 = 8.88308889

C11 = 1188.28051

C12 = 1813.04613

B21 = 0.0483019

B22 = 4.35656721

C21 = 2897.01927

C22 = 5987.80407

References

1(1,2)

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Examples

Estimate the heat capacity of n-decane gas in J/kg/K:

>>> Lastovka_Shaw(1000.0, 0.22491)
3730.2807601773725

Estimate the heat capacity of n-decane gas in J/mol/K:

>>> Lastovka_Shaw(1000.0, 0.22491, MW=142.28)
530.7443465580366

chemicals.heat_capacity.Lastovka_Shaw_integral(T, similarity_variable, cyclic_aliphatic=False, MW=None, term_A=None)

Calculate the integral of ideal-gas constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

Parameters

Tfloat

Temperature of gas [K]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

MWfloat, optional

Molecular weight, [g/mol]

term_Afloat, optional

Term A in Lastovka-Shaw equation, [J/g]

Returns

Hfloat

Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise

See also

Lastovka_Shaw

Lastovka_Shaw_integral_over_T

Notes

Original model is in terms of J/g/K. Integral was computed with SymPy.

References

1

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Examples

>>> Lastovka_Shaw_integral(300.0, 0.1333)
5283095.816018478

chemicals.heat_capacity.Lastovka_Shaw_integral_over_T(T, similarity_variable, cyclic_aliphatic=False, MW=None, term_A=None)

Calculate the integral over temperature of ideal-gas constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

Parameters

Tfloat

Temperature of gas [K]

similarity_variablefloat

Similarity variable as defined in [1], [mol/g]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

MWfloat, optional

Molecular weight, [g/mol]

term_Afloat, optional

Term A in Lastovka-Shaw equation, [J/g]

Returns

Sfloat

Difference in entropy from 0 K, [J/mol/K if MW given; J/kg/K otherwise]

See also

Lastovka_Shaw

Lastovka_Shaw_integral

Notes

Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods! Integral was computed with SymPy.

References

1(1,2)

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Examples

>>> Lastovka_Shaw_integral_over_T(300.0, 0.1333)
3609.791928945323

chemicals.heat_capacity.Lastovka_Shaw_T_for_Hm(Hm, MW, similarity_variable, T_ref=298.15, factor=1.0, cyclic_aliphatic=None, term_A=None)

Uses the Lastovka-Shaw ideal-gas heat capacity correlation to solve for the temperature which has a specified Hm, as is required in PH flashes, as shown in [1].

Parameters

Hmfloat

Molar enthalpy spec, [J/mol]

MWfloat

Molecular weight of the pure compound or mixture average, [g/mol]

similarity_variablefloat

Similarity variable as defined in [1], [mol/g]

T_reffloat, optional

Reference enthlapy temperature, [K]

factorfloat, optional

A factor to increase or decrease the predicted value of the method, [-]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

term_Afloat, optional

Term A in Lastovka-Shaw equation, [J/g]

Returns

Tfloat

Temperature of gas to meet the molar enthalpy spec, [K]

See also

Lastovka_Shaw

Lastovka_Shaw_integral

Lastovka_Shaw_integral_over_T

References

1(1,2)

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Examples

>>> Lastovka_Shaw_T_for_Hm(Hm=55000, MW=80.0, similarity_variable=0.23)
600.0943429567602

chemicals.heat_capacity.Lastovka_Shaw_T_for_Sm(Sm, MW, similarity_variable, T_ref=298.15, factor=1.0, cyclic_aliphatic=None, term_A=None)

Uses the Lastovka-Shaw ideal-gas heat capacity correlation to solve for the temperature which has a specified Sm, as is required in PS flashes, as shown in [1].

Parameters

Smfloat

Molar entropy spec, [J/mol/K]

MWfloat

Molecular weight of the pure compound or mixture average, [g/mol]

similarity_variablefloat

Similarity variable as defined in [1], [mol/g]

T_reffloat, optional

Reference enthlapy temperature, [K]

factorfloat, optional

A factor to increase or decrease the predicted value of the method, [-]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

term_Afloat, optional

Term A in Lastovka-Shaw equation, [J/g]

Returns

Tfloat

Temperature of gas to meet the molar entropy spec, [K]

See also

Lastovka_Shaw

Lastovka_Shaw_integral

Lastovka_Shaw_integral_over_T

References

1(1,2)

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Examples

>>> Lastovka_Shaw_T_for_Sm(Sm=112.80, MW=72.151, similarity_variable=0.2356)
603.4298291570276

chemicals.heat_capacity.Lastovka_Shaw_term_A(similarity_variable, cyclic_aliphatic)

Return Term A in Lastovka-Shaw equation.

Parameters

similarity_variablefloat

Similarity variable as defined in [1], [mol/g]

cyclic_aliphatic: bool, optional

Whether or not chemical is cyclic aliphatic, [-]

Returns

term_Afloat

Term A in Lastovka-Shaw equation, [J/g]

See also

Lastovka_Shaw

Lastovka_Shaw_integral

Lastovka_Shaw_integral_over_T

References

1

Lastovka, Vaclav, and John M. Shaw. “Predictive Correlations for Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions.” Fluid Phase Equilibria 356 (October 25, 2013): 338-370. doi:10.1016/j.fluid.2013.07.023.

Gas Heat Capacity Theory

chemicals.heat_capacity.Cpg_statistical_mechanics(T, thetas, linear=False)

Calculates the ideal-gas heat capacity using of a molecule using its characteristic temperatures, themselves calculated from each of the frequencies of vibration of the molecule. These can be obtained from spectra or quantum mechanical calculations.

$$\frac{C_p^{0}}{R} = \frac{C_p^{0}}{R} \text{rotational} + \frac{C_p^{0}}{R} \text{translational} + \frac{C_p^{0}}{R} \text{vibrational}$$

$$\frac{C_p^{0}}{R} \text{rotational} = 2.5$$

$$\frac{C_p^{0}}{R} \text{translational} = 1 \text{ if linear else } 1.5$$

$$\frac{C_p^{0}}{R} \text{vibrational} = \sum_{i=1}^{3n_A-6+\delta} \left(\frac{\theta_i}{T}\right)^2 \left[\frac{\exp(\theta_i/T)} {\left(\exp(\theta_i/T)-1\right)^2}\right]$$

In the above equation, $delta$ is 1 if the molecule is linear otherwise 0.

Parameters

Tfloat

Temperature of fluid [K]

thetaslist[float]

Characteristic temperatures, [K]

Returns

Cpgmfloat

Gas molar heat capacity at specified temperature, [J/mol/K]

Notes

This equation implies that there is a maximum heat capacity for an ideal gas, and all diatomic or larger gases

Monoatomic gases have a simple heat capacity of 2.5R, the lower limit for ideal gas heat capacity. This function does not cover that type of a gas. The specific gases that are monoatomic are helium, neon, argon, krypton, xenon, radon.

At very low temperatures hydrogen behaves like a monoatomic gas as well.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

Sample calculation in [1] for ammonia:

>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics(300.0, thetas)
35.55983440173097

chemicals.heat_capacity.Cpg_statistical_mechanics_integral(T, thetas, linear=False)

Calculates the integral of ideal-gas heat capacity using of a molecule using its characteristic temperatures.

$$\int {C_p^{0}} = 2.5RT + RT \text{ if linear else } 1.5RT + \int C_p^{0}\text{vibrational}$$

$$\int {C_p^{0}} \text{vibrational} = R\sum_{i=1}^{3n_A-6+\delta} \frac{\theta_i}{\exp(\theta_i/T)-1}$$

Parameters

Tfloat

Temperature of fluid [K]

thetaslist[float]

Characteristic temperatures, [K]

Returns

Hfloat

Integrated gas molar heat capacity at specified temperature, [J/mol]

Examples

>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics_integral(300.0, thetas)
10116.6053294

chemicals.heat_capacity.Cpg_statistical_mechanics_integral_over_T(T, thetas, linear=False)

Calculates the integral over T of ideal-gas heat capacity using of a

molecule using its characteristic temperatures.

$$\int \frac{C_p^{0}}{T} = 2.5R\log(T) + 1R\log(T) \text{ if linear else } 1.5R\log(T) + \int \frac{C_p^{0}}{T}\text{vibrational}$$

$$\int \frac{C_p^{0}}{T} \text{vibrational} = \sum_{i=1}^{3n_A-6+\delta} \frac{\theta_i}{T\exp(\theta_i/T)-T} - \log(\exp(\theta_i/T)-1) + \theta_i/T$$

Parameters

Tfloat

Temperature of fluid [K]

thetaslist[float]

Characteristic temperatures, [K]

Returns

Sfloat

Entropy integral of gas molar heat capacity at specified temperature, [J/mol/K]

Examples

>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics_integral_over_T(300.0, thetas)
190.25658088

chemicals.heat_capacity.vibration_frequency_cm_to_characteristic_temperature(frequency, scale=1)

Convert a vibrational frequency in units of 1/cm to a characteristic temperature for use in calculating heat capacity.

$$\theta = \frac{100\cdot h\cdot c\cdot \text{scale}}{k}$$

Parameters

frequencyfloat

Vibrational frequency, [1/cm]

scalefloat

A scale factor used to adjust the frequency for differences in experimental vs. calculated values, [-]

Returns

thetafloat

Characteristic temperature [K]

Notes

In the equation, k is Boltzmann’s constant, c is the speed of light, and h is the Planck constant.

A scale factor for the MP2/6-31G** method recommended by NIST is 0.9365. Using this scale factor will not improve results in all cases however.

Examples

>>> vibration_frequency_cm_to_characteristic_temperature(667)
959.6641613636505

Liquid Heat Capacity Model Equations

chemicals.heat_capacity.Zabransky_quasi_polynomial(T, Tc, a1, a2, a3, a4, a5, a6)

Calculates liquid heat capacity using the model developed in [1].

$$\frac{C}{R}=A_1\ln(1-T_r) + \frac{A_2}{1-T_r} + \sum_{j=0}^m A_{j+3} T_r^j$$

Parameters

Tfloat

Temperature [K]

Tcfloat

Critical temperature of fluid, [K]

a1-a6float

Coefficients

Returns

Cpfloat

Liquid heat capacity, [J/mol/K]

Notes

Used only for isobaric heat capacities, not saturation heat capacities. Designed for reasonable extrapolation behavior caused by using the reduced critical temperature. Used by the authors of [1] when critical temperature was available for the fluid. Analytical integrals are available for this expression.

References

1(1,2)

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> Zabransky_quasi_polynomial(330, 591.79, -3.12743, 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
165.472878778683

chemicals.heat_capacity.Zabransky_quasi_polynomial_integral(T, Tc, a1, a2, a3, a4, a5, a6)

Calculates the integral of liquid heat capacity using the quasi-polynomial model developed in [1].

Parameters

Tfloat

Temperature [K]

a1-a6float

Coefficients

Returns

Hfloat

Difference in enthalpy from 0 K, [J/mol]

Notes

The analytical integral was derived with SymPy; it is a simple polynomial plus some logarithms.

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> H2 = Zabransky_quasi_polynomial_integral(300, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> H1 = Zabransky_quasi_polynomial_integral(200, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> H2 - H1
14662.031376528757

chemicals.heat_capacity.Zabransky_quasi_polynomial_integral_over_T(T, Tc, a1, a2, a3, a4, a5, a6)

Calculates the integral of liquid heat capacity over T using the quasi-polynomial model developed in [1].

Parameters

Tfloat

Temperature [K]

a1-a6float

Coefficients

Returns

Sfloat

Difference in entropy from 0 K, [J/mol/K]

Notes

The analytical integral was derived with Sympy. It requires the Polylog(2,x) function, which is unimplemented in SciPy. A very accurate numerical approximation was implemented as fluids.numerics.polylog2. Relatively slow due to the use of that special function.

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> S2 = Zabransky_quasi_polynomial_integral_over_T(300, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> S1 = Zabransky_quasi_polynomial_integral_over_T(200, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> S2 - S1
59.16999297436473

chemicals.heat_capacity.Zabransky_cubic(T, a1, a2, a3, a4)

Calculates liquid heat capacity using the model developed in [1].

$$\frac{C}{R}=\sum_{j=0}^3 A_{j+1} \left(\frac{T}{100 \text{K}}\right)^j$$

Parameters

Tfloat

Temperature [K]

a1float

Coefficient, [-]

a2float

Coefficient, [-]

a3float

Coefficient, [-]

a4float

Coefficient, [-]

Returns

Cpfloat

Liquid heat capacity, [J/mol/K]

Notes

Most often form used in [1]. Analytical integrals are available for this expression.

References

1(1,2)

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> Zabransky_cubic(298.15, 20.9634, -10.1344, 2.8253, -0.256738)
75.31465144297

chemicals.heat_capacity.Zabransky_cubic_integral(T, a1, a2, a3, a4)

Calculates the integral of liquid heat capacity using the model developed in [1].

Parameters

Tfloat

Temperature [K]

a1float

Coefficient, [-]

a2float

Coefficient, [-]

a3float

Coefficient, [-]

a4float

Coefficient, [-]

Returns

Hfloat

Difference in enthalpy from 0 K, [J/mol]

Notes

The analytical integral was derived with Sympy; it is a simple polynomial.

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> Zabransky_cubic_integral(298.15, 20.9634, -10.1344, 2.8253, -0.256738)
31051.690370364

chemicals.heat_capacity.Zabransky_cubic_integral_over_T(T, a1, a2, a3, a4)

Calculates the integral of liquid heat capacity over T using the model developed in [1].

Parameters

Tfloat

Temperature [K]

a1float

Coefficient, [-]

a2float

Coefficient, [-]

a3float

Coefficient, [-]

a4float

Coefficient, [-]

Returns

Sfloat

Difference in entropy from 0 K, [J/mol/K]

Notes

The analytical integral was derived with Sympy; it is a simple polynomial, plus a logarithm

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Examples

>>> Zabransky_cubic_integral_over_T(298.15, 20.9634, -10.1344, 2.8253,
... -0.256738)
24.732465342840

class chemicals.heat_capacity.ZabranskySpline(coeffs, Tmin, Tmax)

Implementation of the cubic spline method presented in [1] for calculating the heat capacity of a chemical. Implements the enthalpy and entropy integrals as well.

$$\frac{C}{R}=\sum_{j=0}^3 A_{j+1} \left(\frac{T}{100}\right)^j$$

Parameters

coeffslist[float]

Six coefficients for the equation, [-]

Tminfloat

Minimum temperature any experimental data was available at, [K]

Tmaxfloat

Maximum temperature any experimental data was available at, [K]

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Methods

calculate(T)	Return heat capacity as a function of temperature.
calculate_integral(Ta, Tb)	Return the enthalpy integral of heat capacity from Ta to Tb.
calculate_integral_over_T(Ta, Tb)	Return the entropy integral of heat capacity from Ta to Tb.

calculate(T)

Return heat capacity as a function of temperature.

Parameters

Tfloat

Temperature, [K]

Returns

Cpfloat

Liquid heat capacity as T, [J/mol/K]

calculate_integral(Ta, Tb)

Return the enthalpy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dHfloat

Enthalpy difference between Ta and Tb, [J/mol]

calculate_integral_over_T(Ta, Tb)

Return the entropy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dSfloat

Entropy difference between Ta and Tb, [J/mol/K]

class chemicals.heat_capacity.ZabranskyQuasipolynomial(coeffs, Tc, Tmin, Tmax)

Quasi-polynomial object for calculating the heat capacity of a chemical. Implements the enthalpy and entropy integrals as well.

$$\frac{C}{R}=A_1\ln(1-T_r) + \frac{A_2}{1-T_r} + \sum_{j=0}^m A_{j+3} T_r^j$$

Parameters

coeffslist[float]

Six coefficients for the equation, [-]

Tcfloat

Critical temperature of the chemical, as used in the formula, [K]

Tminfloat

Minimum temperature any experimental data was available at, [K]

Tmaxfloat

Maximum temperature any experimental data was available at, [K]

References

1

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

Methods

calculate(T)	Return the heat capacity as a function of temperature.
calculate_integral(Ta, Tb)	Return the enthalpy integral of heat capacity from Ta to Tb.
calculate_integral_over_T(Ta, Tb)	Return the entropy integral of heat capacity from Ta to Tb.

calculate(T)

Return the heat capacity as a function of temperature.

Parameters

Tfloat

Temperature, [K]

Returns

Cpfloat

Liquid heat capacity as T, [J/mol/K]

calculate_integral(Ta, Tb)

Return the enthalpy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dHfloat

Enthalpy difference between Ta and Tb, [J/mol]

calculate_integral_over_T(Ta, Tb)

Return the entropy integral of heat capacity from Ta to Tb.

Parameters

Tafloat

Initial temperature, [K]

Tbfloat

Final temperature, [K]

Returns

dSfloat

Entropy difference between Ta and Tb, [J/mol/K]

chemicals.heat_capacity.PPDS15(T, Tc, a0, a1, a2, a3, a4, a5)

Calculates the saturation liquid heat capacity using the [1] emperical (parameter-regressed) method, called the PPDS 15 equation for heat capacity.

$$\frac{C_{p,l}}{R} = \frac{a_0}{\tau} + a_1 + a_2\tau + a_3\tau^2 + a_4\tau^3 + a_5\tau^4$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

a0float

Regression parameter, [-]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

a3float

Regression parameter, [-]

a4float

Regression parameter, [-]

a5float

Regression parameter, [-]

Returns

Cplmfloat

Liquid molar saturation heat capacity, [J/mol/K]

References

1(1,2)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-CsatL/PPDS15-Csat.htm.

Examples

Benzene at 400 K from [1]

>>> PPDS15(T=400.0, Tc=562.05, a0=0.198892, a1=24.1389, a2=-20.2301, a3=5.72481, a4=4.43613e-7, a5=-3.10751e-7)
161.8983143509

chemicals.heat_capacity.TDE_CSExpansion(T, Tc, b, a1, a2=0.0, a3=0.0, a4=0.0)

Calculates the saturation liquid heat capacity using the [1] CSExpansion method from NIST’s TDE:

$$C_{p,l}= \frac{b}{\tau} + a_1 + a_2T + a_3 T^2 + a_4 T^3$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

bfloat

Regression parameter, [-]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

a3float

Regression parameter, [-]

a4float

Regression parameter, [-]

Returns

Cplmfloat

Liquid molar saturation heat capacity, [J/mol/K]

References

1(1,2)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-CsatL/CSExpansion.htm

Examples

2-methylquinoline at 550 K from [1]

>>> TDE_CSExpansion(550.0, 778.0, 0.626549, 120.705, 0.255987, 0.000381027, -3.03077e-7)
328.472042686

Liquid Heat Capacity Estimation Models

chemicals.heat_capacity.Rowlinson_Poling(T, Tc, omega, Cpgm)

Calculate liquid constant-pressure heat capacity with the [1] CSP method. This equation is not terrible accurate.

The heat capacity of a liquid is given by:

$$\frac{Cp^{L} - Cp^{g}}{R} = 1.586 + \frac{0.49}{1-T_r} + \omega\left[ 4.2775 + \frac{6.3(1-T_r)^{1/3}}{T_r} + \frac{0.4355}{1-T_r}\right]$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

omegafloat

Acentric factor for fluid, [-]

Cpgmfloat

Constant-pressure gas heat capacity, [J/mol/K]

Returns

Cplmfloat

Liquid constant-pressure heat capacity, [J/mol/K]

Notes

Poling compared 212 substances, and found error at 298K larger than 10% for 18 of them, mostly associating. Of the other 194 compounds, AARD is 2.5%.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

Examples

>>> Rowlinson_Poling(350.0, 435.5, 0.203, 91.21)
143.80196224081436

chemicals.heat_capacity.Rowlinson_Bondi(T, Tc, omega, Cpgm)

Calculate liquid constant-pressure heat capacity with the CSP method shown in [1].

The heat capacity of a liquid is given by:

$$\frac{Cp^L - Cp^{ig}}{R} = 1.45 + 0.45(1-T_r)^{-1} + 0.25\omega [17.11 + 25.2(1-T_r)^{1/3}T_r^{-1} + 1.742(1-T_r)^{-1}]$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

omegafloat

Acentric factor for fluid, [-]

Cpgmfloat

Constant-pressure gas heat capacity, [J/mol/K]

Returns

Cplmfloat

Liquid constant-pressure heat capacity, [J/mol/K]

Notes

Less accurate than Rowlinson_Poling.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

2

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

3

J.S. Rowlinson, Liquids and Liquid Mixtures, 2nd Ed., Butterworth, London (1969).

Examples

>>> Rowlinson_Bondi(T=373.28, Tc=535.55, omega=0.323, Cpgm=119.342)
175.3976263003074

chemicals.heat_capacity.Dadgostar_Shaw(T, similarity_variable, MW=None, terms=None)

Calculate liquid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

$$C_{p} = 24.5(a_{11}\alpha + a_{12}\alpha^2)+ (a_{21}\alpha + a_{22}\alpha^2)T +(a_{31}\alpha + a_{32}\alpha^2)T^2$$

Parameters

Tfloat

Temperature of liquid [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

termsfloat, optional

Terms in Dadgostar-Shaw equation as computed by Dadgostar_Shaw_terms

Returns

Cplfloat

Liquid constant-pressure heat capacity, J/mol/K if MW given; J/kg/K otherwise

Notes

Many restrictions on its use. Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods! a11 = -0.3416; a12 = 2.2671; a21 = 0.1064; a22 = -0.3874l; a31 = -9.8231E-05; a32 = 4.182E-04

References

1(1,2)

Dadgostar, Nafiseh, and John M. Shaw. “A Predictive Correlation for the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined Liquid Hydrocarbons.” Fluid Phase Equilibria 313 (January 15, 2012): 211-226. doi:10.1016/j.fluid.2011.09.015.

Examples

>>> Dadgostar_Shaw(355.6, 0.139)
1802.5291501191516

chemicals.heat_capacity.Dadgostar_Shaw_integral(T, similarity_variable, MW=None, terms=None)

Calculate the integral of liquid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

Parameters

Tfloat

Temperature of gas [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

termsfloat, optional

Terms in Dadgostar-Shaw equation as computed by Dadgostar_Shaw_terms

Returns

Hfloat

Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise

See also

Dadgostar_Shaw

Dadgostar_Shaw_integral_over_T

Notes

Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods! Integral was computed with SymPy.

References

1(1,2)

Dadgostar, Nafiseh, and John M. Shaw. “A Predictive Correlation for the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined Liquid Hydrocarbons.” Fluid Phase Equilibria 313 (January 15, 2012): 211-226. doi:10.1016/j.fluid.2011.09.015.

Examples

>>> Dadgostar_Shaw_integral(300.0, 0.1333)
238908.15142664989

chemicals.heat_capacity.Dadgostar_Shaw_integral_over_T(T, similarity_variable, MW=None, terms=None)

Calculate the integral of liquid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

Parameters

Tfloat

Temperature of gas [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

termsfloat, optional

Terms in Dadgostar-Shaw equation as computed by Dadgostar_Shaw_terms

Returns

Sfloat

Difference in entropy from 0 K, J/mol/K if MW given; J/kg/K otherwise

See also

Dadgostar_Shaw

Dadgostar_Shaw_integral

Notes

Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods! Integral was computed with SymPy.

References

1(1,2)

Dadgostar, Nafiseh, and John M. Shaw. “A Predictive Correlation for the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined Liquid Hydrocarbons.” Fluid Phase Equilibria 313 (January 15, 2012): 211-226. doi:10.1016/j.fluid.2011.09.015.

Examples

>>> Dadgostar_Shaw_integral_over_T(300.0, 0.1333)
1201.1409113147918

chemicals.heat_capacity.Dadgostar_Shaw_terms(similarity_variable)

Return terms for the computation of Dadgostar-Shaw heat capacity equation.

Parameters

similarity_variablefloat

Similarity variable, [mol/g]

Returns

firstfloat

First term, [-]

secondfloat

Second term, [-]

thirdfloat

Third term, [-]

See also

Dadgostar_Shaw

Solid Heat Capacity Estimation Models

chemicals.heat_capacity.Perry_151(T, a, b, c, d)

Return the solid molar heat capacity of a chemical using the Perry 151 method, as described in [1].

Parameters

a,b,c,dfloat

Regressed coefficients.

Returns

Cpsfloat

Solid constant-pressure heat capacity, [J/mol/K]

Notes

The solid heat capacity is given by:

$$C_n = 4.184 (a + bT + \frac{c}{T^2} + dT^2)$$

Coefficients are listed in section 2, table 151 of [1]. Note that the original model was in a Calorie basis, but has been translated to Joules.

References

1(1,2)

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

Heat capacity of solid aluminum at 300 K:

>>> Perry_151(300, 4.8, 0.00322, 0., 0.)
24.124944

chemicals.heat_capacity.Lastovka_solid(T, similarity_variable, MW=None)

Calculate solid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

$$C_p = 3(A_1\alpha + A_2\alpha^2)R\left(\frac{\theta}{T}\right)^2 \frac{\exp(\theta/T)}{[\exp(\theta/T)-1]^2} + (C_1\alpha + C_2\alpha^2)T + (D_1\alpha + D_2\alpha^2)T^2$$

Parameters

Tfloat

Temperature of solid [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

Returns

Cpsfloat

Solid constant-pressure heat capacity, J/mol/K if MW given; J/kg/K otherwise

Notes

Many restrictions on its use. Trained on data with MW from 12.24 g/mol to 402.4 g/mol, C mass fractions from 61.3% to 95.2%, H mass fractions from 3.73% to 15.2%, N mass fractions from 0 to 15.4%, O mass fractions from 0 to 18.8%, and S mass fractions from 0 to 29.6%. Recommended for organic compounds with low mass fractions of hetero-atoms and especially when molar mass exceeds 200 g/mol. This model does not show and effects of phase transition but should not be used passed the triple point. Original model is in terms of J/g/K. Note that the model s for predicting mass heat capacity, not molar heat capacity like most other methods!

A1 = 0.013183

A2 = 0.249381

$\theta$ = 151.8675

C1 = 0.026526

C2 = -0.024942

D1 = 0.000025

D2 = -0.000123

References

1(1,2)

Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw. “A Similarity Variable for Estimating the Heat Capacity of Solid Organic Compounds: Part II. Application: Heat Capacity Calculation for Ill-Defined Organic Solids.” Fluid Phase Equilibria 268, no. 1-2 (June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.

Examples

>>> Lastovka_solid(300, 0.2139)
1682.0637469909211

chemicals.heat_capacity.Lastovka_solid_integral(T, similarity_variable, MW=None)

Integrates solid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

uses an explicit form as derived with Sympy.

Parameters

Tfloat

Temperature of solid [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

Returns

Hfloat

Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise

See also

Lastovka_solid

Notes

Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods!

References

1(1,2)

Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw. “A Similarity Variable for Estimating the Heat Capacity of Solid Organic Compounds: Part II. Application: Heat Capacity Calculation for Ill-Defined Organic Solids.” Fluid Phase Equilibria 268, no. 1-2 (June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.

Examples

>>> Lastovka_solid_integral(300, 0.2139)
283246.1519409122

chemicals.heat_capacity.Lastovka_solid_integral_over_T(T, similarity_variable, MW=None)

Integrates over T solid constant-pressure heat capacity with the similarity variable concept and method as shown in [1].

uses an explicit form as derived with Sympy.

Parameters

Tfloat

Temperature of solid [K]

similarity_variablefloat

similarity variable as defined in [1], [mol/g]

MWfloat, optional

Molecular weight of the pure compound or mixture average, [g/mol]

Returns

Sfloat

Difference in entropy from 0 K, J/mol/K if MW given; J/kg/K otherwise

See also

Lastovka_solid

Notes

Original model is in terms of J/g/K. Note that the model is for predicting mass heat capacity, not molar heat capacity like most other methods!

References

1(1,2)

Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw. “A Similarity Variable for Estimating the Heat Capacity of Solid Organic Compounds: Part II. Application: Heat Capacity Calculation for Ill-Defined Organic Solids.” Fluid Phase Equilibria 268, no. 1-2 (June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.

Examples

>>> Lastovka_solid_integral_over_T(300, 0.2139)
1947.5537561495564

Utility methods

class chemicals.heat_capacity.PiecewiseHeatCapacity(models)

Create a PiecewiseHeatCapacity object for calculating heat capacity and the enthalpy and entropy integrals using piecewise models.

Parameters

modelsIterable[HeatCapacity]

Piecewise heat capacity objects, [-]

Attributes

Tmax

Tmin

models

Methods

calculate(T)	Return the heat capacity as a function of temperature.
calculate_integral(Ta, Tb)	Return the enthalpy integral of heat capacity from Ta to Tb.
calculate_integral_over_T(Ta, Tb)	Return the entropy integral of heat capacity from Ta to Tb.
force_calculate(T)	Return the heat capacity as a function of temperature.
force_calculate_integral(Ta, Tb)	Return the enthalpy integral of heat capacity from Ta to Tb.
force_calculate_integral_over_T(Ta, Tb)	Return the entropy integral of heat capacity from Ta to Tb.

Fit Coefficients

All of these coefficients are lazy-loaded, so they must be accessed as an attribute of this module.

chemicals.heat_capacity.Cp_data_Poling

Constains data for gases and liquids from [3]. Simple polynomials for gas heat capacity (not suitable for extrapolation) are available for 308 chemicals. Additionally, constant values in at 298.15 K are available for 348 gases. Constant values in at 298.15 K are available for 245 liquids.

chemicals.heat_capacity.TRC_gas_data

A rigorous expression from [1] for modeling gas heat capacity. Coefficients for 1961 chemicals are available.

chemicals.heat_capacity.CRC_standard_data

Constant values tabulated in [4] at 298.15 K. Data is available for 533 gases. Data is available for 433 liquids. Data is available for 529 solids.

chemicals.heat_capacity.Cp_dict_PerryI

Simple polynomials from [5] with vaious exponents selected for each expression. Coefficients are in units of calories/mol/K. The full expression is $C_p = a + bT + c/T^2 + dT^2$. Data is available for 284 compounds. Some compounds have gas data, some have liquid data, and have solid (crystal structure) data, sometimes multiple coefficients for different solid phases.

chemicals.heat_capacity.zabransky_dicts

Complicated fits covering different cases and with different forms from [2].

chemicals.heat_capacity.Cp_dict_characteristic_temperatures_adjusted_psi4_2022a

Theoretically calculated chatacteristic temperatures from vibrational frequencies using psi4

chemicals.heat_capacity.Cp_dict_characteristic_temperatures_psi4_2022a

Theoretically calculated chatacteristic temperatures from vibrational frequencies using psi4, adjusted using a recommended coefficient

1

Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.

2

Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski. Heat Capacity of Liquids: Critical Review and Recommended Values. 2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.

3

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

4

Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.

5

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

In [1]: import chemicals

In [2]: chemicals.heat_capacity.Cp_data_Poling
Out[2]: 
                                  Chemical   Tmin  ...     Cpg     Cpl
CAS                                                ...                
56-23-5                tetrachloromethane   200.0  ...   83.43  131.60
60-29-7                     diethyl ether   100.0  ...  119.46  172.60
62-53-3            benzeneamine (aniline)    50.0  ...  107.90  191.90
64-17-5                           ethanol    50.0  ...   65.21  112.25
64-18-6      methanoic acid (formic acid)    50.0  ...   53.45   99.17
...                                    ...    ...  ...     ...     ...
14940-65-9                  tritium oxide     NaN  ...   34.96     NaN
16747-38-9     2,3,3,4-tetramethylpentane   200.0  ...  218.30  275.70
20291-95-6         2,2,5-trimethylheptane   200.0  ...  229.20  306.40
800000-51-5              hydrogen, normal     NaN  ...   28.83     NaN
800000-54-8             deuterium, normal     NaN  ...   29.20     NaN

[368 rows x 10 columns]

In [3]: chemicals.heat_capacity.TRC_gas_data
Out[3]: 
                                    Chemical   Tmin  ...      J       Hfg
CAS                                                  ...                 
50-00-0                             Methanal   50.0  ...   3.46 -104700.0
50-32-8                       Benzo[a]pyrene  298.0  ...  13.44  324000.0
53-70-3                Dibenz[a,h]anthracene  298.0  ...  16.63  375000.0
56-23-5                   Tetrachloromethane  200.0  ...   9.58  -93700.0
56-55-3                    Benz[a]anthracene  298.0  ...  11.45  328000.0
...                                      ...    ...  ...    ...       ...
800000-46-8  2,2,(3RS,4RS)-Tetramethylhexane  200.0  ...  22.45 -188600.0
800000-47-9  2,(3RS,4SR),5-Tetramethylhexane  200.0  ...  22.32  193700.0
800000-48-0  2,(3RS,4RS),5-Tetramethylhexane  200.0  ...  22.14 -194600.0
800000-56-0            1-Methylbutyl radical  200.0  ...  22.25   54600.0
800002-32-8           Propenoic acid (Dimer)   50.0  ...  13.83 -686000.0

[1961 rows x 14 columns]

In [4]: chemicals.heat_capacity.CRC_standard_data
Out[4]: 
                                Chemical        Hfs  ...    S0g    Cpg
CAS                                                  ...              
50-00-0                     Formaldehyde        NaN  ...  218.8   35.4
50-32-8                   Benzo[a]pyrene        NaN  ...    NaN  254.8
50-69-1                         D-Ribose -1047200.0  ...    NaN    NaN
50-78-2        2-(Acetyloxy)benzoic acid  -815600.0  ...    NaN    NaN
50-81-7                  L-Ascorbic acid -1164600.0  ...    NaN    NaN
...                                  ...        ...  ...    ...    ...
92141-86-1             Cesium metaborate  -972000.0  ...    NaN    NaN
99685-96-8        Carbon [fullerene-C60]  2327000.0  ...  544.0  512.0
114489-96-2  Isobutyl 2-chloropropanoate        NaN  ...    NaN    NaN
115383-22-7       Carbon [fullerene-C70]  2555000.0  ...  614.0  585.0
116836-32-9         sec-Butyl pentanoate        NaN  ...    NaN    NaN

[2470 rows x 13 columns]

In [5]: chemicals.heat_capacity.Cp_dict_PerryI['124-38-9'] # gas only
Out[5]: 
{'g': {'Formula': 'CO2',
  'Phase': 'g',
  'Subphase': None,
  'Const': 10.34,
  'Lin': 0.00274,
  'Quadinv': -195500.0,
  'Quad': 0,
  'Tmin': 273.0,
  'Tmax': 1200.0,
  'Error': '1a'}}

In [6]: chemicals.heat_capacity.Cp_dict_PerryI['7704-34-9'] # crystal and gas
Out[6]: 
{'g': {'Formula': 'H2S',
  'Phase': 'g',
  'Subphase': None,
  'Const': 7.2,
  'Lin': 0.0036,
  'Quadinv': 0,
  'Quad': 0,
  'Tmin': 300.0,
  'Tmax': 600.0,
  'Error': 8.0},
 'c': {'Formula': 'S',
  'Phase': 'c',
  'Subphase': 'monoclinic',
  'Const': 4.38,
  'Lin': 0.0044,
  'Quadinv': 0,
  'Quad': 0,
  'Tmin': 368.0,
  'Tmax': 392.0,
  'Error': 3.0}}

In [7]: chemicals.heat_capacity.Cp_dict_PerryI['7440-57-5'] # crystal and liquid
Out[7]: 
{'c': {'Formula': 'Au',
  'Phase': 'c',
  'Subphase': None,
  'Const': 5.61,
  'Lin': 0.00144,
  'Quadinv': 0,
  'Quad': 0,
  'Tmin': 273.0,
  'Tmax': 1336.0,
  'Error': 2.0},
 'l': {'Formula': 'Au',
  'Phase': 'l',
  'Subphase': None,
  'Const': 7.0,
  'Lin': 0,
  'Quadinv': 0,
  'Quad': 0,
  'Tmin': 1336.0,
  'Tmax': 1573.0,
  'Error': 5.0}}

In [8]: chemicals.heat_capacity.zabransky_dicts.keys()
Out[8]: dict_keys(['Zabransky spline, averaged heat capacity', 'Zabransky quasipolynomial, averaged heat capacity', 'Zabransky spline, constant-pressure', 'Zabransky quasipolynomial, constant-pressure', 'Zabransky spline, saturation', 'Zabransky quasipolynomial, saturation'])