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: float, a0: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float, a7: float) → float

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

$$C_p = R\left(a_0 + (a_1/T^2) \exp(-a_2/T) + a_3 y^2 + (a_4 - a_5/(T-a_7)^2 )y^8 \right)$$

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

Parameters:

Tfloat

Temperature [K]

a0float

Constant coefficient, [-]

a1float

a1 coefficient, [K^2]

a2float

a2 coefficient, [K]

a3float

a3 coefficient, [-]

a4float

a4 coefficient, [-]

a5float

a5 coefficient, [K^2]

a6float

a6 coefficient, [K]

a7float

a6 coefficient, [K]

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: float, a0: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float, a7: float, I: float = 0) → float

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]

a0float

Constant coefficient, [-]

a1float

a1 coefficient, [K^2]

a2float

a2 coefficient, [K]

a3float

a3 coefficient, [-]

a4float

a4 coefficient, [-]

a5float

a5 coefficient, [K^2]

a6float

a6 coefficient, [K]

a7float

a6 coefficient, [K]

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: float, a0: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float, a7: float, J: float = 0) → float

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:

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

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

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

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

Parameters:

Tfloat

Temperature [K]

a0float

Constant coefficient, [-]

a1float

a1 coefficient, [K^2]

a2float

a2 coefficient, [K]

a3float

a3 coefficient, [-]

a4float

a4 coefficient, [-]

a5float

a5 coefficient, [K^2]

a6float

a6 coefficient, [K]

a7float

a6 coefficient, [K]

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]

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.

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

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]

afloat

Constant coefficient, [-]

bfloat

Linear coefficient, [-]

cfloat

Quadratic coefficient, [-]

dfloat

Cubic coefficient, [-]

efloat

Quintic coefficient, [=]

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]

afloat

Constant coefficient, [-]

bfloat

Linear coefficient, [-]

cfloat

Quadratic coefficient, [-]

dfloat

Cubic coefficient, [-]

efloat

Quintic coefficient, [=]

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]

afloat

Constant coefficient, [-]

bfloat

Linear coefficient, [-]

cfloat

Quadratic coefficient, [-]

dfloat

Cubic coefficient, [-]

efloat

Quintic coefficient, [=]

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.

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

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

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_aliphaticbool, 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]

similarity_variablefloat

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

cyclic_aliphaticbool, 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] (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(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_aliphaticbool, 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=False)

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_aliphaticbool, optional

Whether or not chemical is cyclic aliphatic, [-]

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=False)

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_aliphaticbool, optional

Whether or not chemical is cyclic aliphatic, [-]

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)
603.429829157

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_aliphaticbool, 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]

linearbool, optional

Whether the molecule is linear or not, [-]

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]

linearbool, optional

Whether the molecule is linear or not, [-]

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]

linearbool, optional

Whether the molecule is linear or not, [-]

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=1.0)

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, optional

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: float, Tc: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float) → float

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]

a1float

First coefficient, [-]

a2float

Second coefficient, [-]

a3float

Third coefficient, [-]

a4float

Fourth coefficient, [-]

a5float

Fifth coefficient, [-]

a6float

Sixth coefficient, [-]

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: float, Tc: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float) → float

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

Parameters:

Tfloat

Temperature [K]

Tcfloat

Critical temperature of fluid, [K]

a1float

First coefficient, [-]

a2float

Second coefficient, [-]

a3float

Third coefficient, [-]

a4float

Fourth coefficient, [-]

a5float

Fifth coefficient, [-]

a6float

Sixth coefficient, [-]

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: float, Tc: float, a1: float, a2: float, a3: float, a4: float, a5: float, a6: float) → float

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

Parameters:

Tfloat

Temperature [K]

Tcfloat

Critical temperature of fluid, [K]

a1float

First coefficient, [-]

a2float

Second coefficient, [-]

a3float

Third coefficient, [-]

a4float

Fourth coefficient, [-]

a5float

Fifth coefficient, [-]

a6float

Sixth coefficient, [-]

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: float, a1: float, a2: float, a3: float, a4: float) → float

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: float, a1: float, a2: float, a3: float, a4: float) → float

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: float, a1: float, a2: float, a3: float, a4: float) → float

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: tuple[float, float, float, float], Tmin: float, Tmax: float)

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]

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.

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.

calculate(T: float) → float

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: float, Tb: float) → float

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: float, Tb: float) → float

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: tuple[float, float, float, float, float, float], Tc: float, Tmin: float, Tmax: float)

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]

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.

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.

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: float, Tc: float, omega: float, Cpgm: float) → float

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: float, Tc: float, omega: float, Cpgm: float) → float

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)

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]

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)

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]

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)

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]

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: list[ZabranskySpline])

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

chemicals.heat_capacity.Cp_data_Perry_Table_153_100

A collection of 333 compound coefficient sets for heat capacity of liquids at various temperatures. The coefficients are in the form of a polynomial of degree 5, which supports the DIPPR equation 100 from [5]. The coefficients are in units of J/kmol/K.

chemicals.heat_capacity.Cp_data_Perry_Table_153_114

A collection of 11 compound coefficient sets for heat capacity of liquids at various temperatures. The coefficients are in the form of a polynomial of degree 5, which supports the DIPPR equation 114 from [5]. The coefficients are in units of J/kmol/K.

[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] (1,2,3)

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

The structure of each dataframe is shown below:

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
2099474000-00-0              hydrogen, normal     NaN  ...   28.83     NaN
2099437000-00-0             deuterium, normal     NaN  ...   29.20     NaN

[367 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'])

In [9]: chemicals.heat_capacity.Cp_data_Perry_Table_153_100.loc['75-07-0']
Out[9]: 
Chemical    Acetaldehyde
A               115100.0
B                 -433.0
C                  1.425
D                    0.0
E                    0.0
Tmin              150.15
Tmax               294.0
Name: 75-07-0, dtype: object

In [10]: chemicals.heat_capacity.Cp_data_Perry_Table_153_114.loc['7664-41-7']
Out[10]: 
Name    Ammonia
A        61.289
B       80925.0
C         799.4
D       -2651.0
E             0
Tmin     203.15
Tmax     401.15
Name: 7664-41-7, dtype: object