### Category: Properties

Property Estimation Joback Method

## Property Estimation Joback Method

The Joback method is a group contribution method which calculates thermophysical and transport properties as a function of the sum of group parameters. It uses a very simple and easy to assign group scheme.

Example
Estimate properties for 1-Butanol based on Joback Method

Structure of 1-Butanol consists of following groups

````-CH3 Group : 1`
`-CH2 Group : 3`
`-OH (alcohol) Group : 1````

Normal Boiling Point

``TNBP (K) = 198 + ΣTb,i``

where Tb,i is contribution due to each group. These values for individual group are available in literature and Wikipedia reference below. On combining values for each group normal boiling point comes out to be –

``TNBP (K) = 383.10``

Critical Temperature

````Tc (K) = TNBP /(0.584 + 0.965*ΣTc,i - (ΣTc,i)²)`
`Tc (K) = 545.08````

Critical Pressure

````Pc (bar) = (0.113 + 0.0032*NA - ΣPc,i)-2`
`Pc (bar) = 43.86````

where NA is number of atoms in the molecular structure.
Critical Volume

````Vc (cm³/mol) = 17.5 + ΣVc,i`
`Vc (cm³/mol) = 278.50````

Critical Compressibility

````Zc = (Pc.Vc)/(R.Tc)`
`Zc = 0.2695````

Acentric Factor
Lee-Kesler method can be used to estimate the acentric factor.

````ω = α/β`
`θ = TNBP / Tc`
`α = -ln(Pc) - 5.92714 + 6.09648/θ + 1.28862.ln(θ) - 0.169347.θ6`
`β = 15.2518 - 15.6875/θ - 13.4721.ln(θ) + 0.43577.θ6`
`ω = 0.6602````

Freezing Point

````Tm (K) = 122.5 + ΣTm,i`
`Tm (K) = 195.66````

Heat of Formation (Ideal Gas, 298 K)

````Hformation (kJ/mol) = 68.29 + ΣHform,i`
`Hformation (kJ/mol) = -278.12````

Gibbs Energy of Formation (Ideal Gas, 298 K)

````Gformation (kJ/mol) = 53.88 + ΣGform,i`
`Gformation (kJ/mol) = -154.02````

Heat of Vaporization (at Normal Boiling Point)
Reidel’s equation can be used to estimate a liquid’s heat of vaporization at its normal boiling point.

````ΔHv (kJ/mol) = 1.092.R.TNBP .(ln(Pc) -1.013)/(0.930 - (TNBP / Tc))`
`ΔHv (kJ/mol) = 42.38````

Heat of Fusion

````ΔHfus (kJ/mol) = -0.88 + ΣHfus,i`
`ΔHfus (kJ/mol) = 10.2````

Heat Capacity (Ideal Gas)

````CP (J/mol.K) = A + B.T + C.T² + D.T³`
`A = Σai - 37.93`
`B = Σbi + 0.210`
`C = Σci - 3.91*10-4`
`D = Σdi + 2.06*10-7`
`CP (J/mol.K) = 110.96 (at 300 K)````

Heat of Vaporization (@ Temperature T)
Watson equation can be used to estimate heat of vaporization at different temperature as following.

````ΔHv,2 (kJ/mol) = Hv,1*((Tc - T2)/(Tc - T1))0.38`
`ΔHv,2 (kJ/mol) = 49.60 (at 300 K)````

Liquid Viscosity

````ηL (Pa.s) = Mw.exp(A/T + B)`
`A = Σηa - 597.82`
`B = Σηb - 11.202`
`ηL (Pa.s) = 1.936*10-3 (at 300 K)````

where Mw is the molecular weight.
Liquid Density
Rackett equation can be used to estimate liquid density from critical properties as following –

````ρL (gm/cm³) = Mw/( (R.Tc/Pc)*Zc^(1 + (1- T/Tc)2/7))`
`ρL (gm/cm³) = 0.756 (at 300 K)````

Liquid Vapor Pressure
Lee-Kesler equation can be used to estimate liquid vapor pressure as following –

````Pvp (bar) = Pc.exp(f0 + ω.f1)`
`f0 = 5.92714 - 6.09648/Tr - 1.28862*ln(Tr) + 0.169347.Tr6`
`f1 = 15.2518 - 15.6875/Tr - 13.4721*ln(Tr) + 0.43577.Tr6`
`Tr = T / Tc`
`Pvp (bar) = 0.018````

Web based calculation available at CheCalc.com

Spreadsheet for property estimation based on Joback Method

### References

Specific Heat Ratio of Real Gas

## Specific Heat Ratio of Real Gas

Equation of state is used to derive variety of thermodynamic properties. This article illustrate calculation of specific heat ratio from Peng Robinson Equation of state.

Example
Calculate specific heat ratio ( γ = Cp/Cv ) for methane gas at 11 Bar & 300 °K. Critical constants for Methane are as following

• Critical temperature, Tc : 190.6°K
• Critical Pressure, Pc : 46.002 bar
• Accentric Factor, ω : 0.008

Ideal gas specific heat constants CpIG = A + B.T + C.T² + D.T³ are as following

• A = 4.5980
• B = 0.0125
• C = 2.86 x 10-6
• D = -2.7 x 10-9

where Cp is in cal/mol-K

Peng Robinson equation of state is defined as

`` P = RT / (V - b)  - a / [V(V + b) + b(V - b)]``

where

```` ac = 0.45723553 R²Tc²/Pc`
` b = 0.077796074 RTc/Pc`
` m = 0.37464 + 1.54226ω - 0.26992ω² `
` a = ac[1 + m(1 - (T/Tc)0.5)]²````

Above equation is translated into polynomial form and solved for values of Z using Newton-Raphson method.

```` Z³ - (1 - B)Z² + Z (A - 2B - 3B²) - (AB - B² - B³) = 0`
` Z = PV/RT`
` A = aP/ (RT)²`
` B = bP/ RT````

Following partial derivatives are required for calculating thermodynamic properties. First derivative is obtained by differentiation of P with respect to V at constant T.

```` (δP/ δV)T = -RT/(v - b)² + 2a(v + b)/[v(v + b) + b(v - b)]²`
` (δP/ δV)T = -0.00485 bar/(cm3/mol)````

Second derivative is obtained by differentiation of P with respect to T at constant V.

```` (δP/ δT)V = R/(v - b) - a'/[v(v + b) + b(v - b)]`
` (δa/ δT)V = -mac/[(TTc)0.5(1 + m( 1 - (T/Tc)0.5))]`
` (δP/ δT)V = 0.039 bar/K`
` (δT/ δP)V = 25.814 K/bar````

Third derivative is obtained by differentiation of V with respect to T at constant P.

```` (δV/ δT)P = (R/P)[ T(δZ/δT)P + Z]`
` (δZ/ δT)P = Num / Denom`
` Num = (δA/δT)P (B-Z) + (δB/δT)P(6BZ+2Z-3B²-2B+A-Z²)`
` Denom = 3Z&sup2 + 2(B-1)Z + (A-2B-3B²)````

where,

```` (δA/δT)P = (P/(RT)²)(a' - 2a/T)`
` (δB/δT)P = -bP/(RT²)````

### Calculation of Heat Capacities

Ideal gas heat capacity CpIG is calculated at 300 °K from polynomial equation provided above. Specific heat at constant volume for ideal gas, CvIG is calculated using following relation.

`` CvIG = CpIG - R``

Residual heat capacity at constant volume Cv R is calculated from internal energy U R as following.

```` CvR = (δUR/δT)V`
` UR = [(Ta'-a)/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]`
` CvR = [Ta"/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]````

where,

`` a" = ac m (1 + m)(Tc/T)0.5/ (2TTc)``

Specific heat capacity at constant pressure and volume is calculated using following equation.

```` CpR = CvR + T(δP/δT)V(δV/δT)P - R`
` Cp = CpIG + CpR`
` Cv = CvIG + CvR````

Specific heat ratio is obtained as :

```` γ = Cp / Cv`
` γ = 1.338````

Spreadsheet for Specific heat ratio calculaton from Peng Robinson EOS