estimation of properties

Chemical Engineering 317: Additional Notes – Properties of Compounds of 34

CHEMICAL ENGINEERING 317

ADDITIONAL NOTES: PROPERTIES OF COMPOUNDS

CONTENTS

1 Methods for estimation of physical properties .............................................................................. 2 1.1 Law of corresponding states ................................................................................................ 2 1.2 Group contribution methods................................................................................................. 2 1.3 (Semi) Empirical correlations ............................................................................................... 3

2 Pure component constants........................................................................................................... 3 2.1 Critical constants (pressure, temperature, volume, compressibility) ................................... 3

⋅ Critical temperature ..................................................................................................... 4 ⋅ Critical pressure .......................................................................................................... 5 ⋅ Critical volume............................................................................................................. 6 ⋅ Critical properties of mixtures...................................................................................... 7

2.2 Normal boiling point ............................................................................................................. 7 2.3 Acentric factor ...................................................................................................................... 8

3 Vapour pressure of liquids............................................................................................................ 9

4 Thermal properties...................................................................................................................... 14 4.1 Heat capacity / enthalpy..................................................................................................... 14

⋅ Vapour heat capacity at constant pressure............................................................... 15 ⋅ Liquid heat capacity at constant pressure................................................................. 16 ⋅ Heat capacity of mixtures.......................................................................................... 17

4.2 Enthalpy of phase changes................................................................................................ 18 ⋅ Enthalpy (heat) of vapourisation (liquid to vapour) ................................................... 18 ⋅ Enthalpy (heat) of fusion (liquid to solid) ................................................................... 20

4.3 Enthalpy (heat) of formation............................................................................................... 20

5 Physical properties ..................................................................................................................... 21

6 Web sources............................................................................................................................... 21

7 Nomenclature ............................................................................................................................. 22

8 References ................................................................................................................................. 23

9 Appendices ................................................................................................................................. 24 9.1 Group contribution parameters for the Lydersen’s method ............................................... 24 9.2 Group contribution parameters for the Constantinou and Gani’s method ......................... 25 9.3 Group increments for the Fedors’ method ......................................................................... 27 9.4 Constants for the Antoine equation ................................................................................... 27 9.5 Parameters for method of Harrison and Seaton ................................................................ 28 9.6 Group contribution parameters for Rihani and Doraiswamy’s method.............................. 29 9.7 Group contribution parameters for Shaw’s method ........................................................... 31 9.8 Group contribution parameters for Růžička and Domalski’s method ................................ 32

Many engineering applications require the use of physical properties. Some of these properties are published in the literature yet in many cases these properties have not been measured yet. Although it may be possible to measure the physical properties of a substance experimentally, these types of experiments are often tedious and time consuming. Thus in the absence of reliable experimental data physical properties need to be estimated with sufficient accuracy for the purpose required. The purpose of this text is not to discuss all property estimation methods in detail but rather to introduce


some of the common methods used to estimate some properties, specifically the properties used in chemical thermodynamics.

1 METHODS FOR ESTIMATION OF PHYSICAL PROPERTIES

Various methods are used in development of the estimation correlations. Some are based on theory with empirical parameters, other totally based on fitting available data while some are based on the theory of corresponding states. Before we discuss the methods to predict some properties, a short discussion on some general methods widely used is presented:

1.1 Law of corresponding states

Proposed by van der Waals in 1873, the law of corresponding states expresses the generalisation that equilibrium properties which depend on intermolecular forces are related to the critical properties in a universal kind of way. The law of corresponding states provides the single most important basis for the development of correlations and estimation methods. The macroscopic version of the law of corresponding states is as follows: All fluids at the same reduced temperature and reduced pressure have the same compressibility factor. Consider, for example, a plot of the reduced pressure versus the reduced volume for methane and nitrogen at saturated conditions and at a reduced temperature of 1.1. (See Figure 1). As seen, the plots coincide well and illustrate the principle of corresponding states.

0.01

0.1

1

10

0.1 1 10 100 1000Reduced Volume

Red

uced

Pre

ssur

e

Methane Saturated

Nitrogen Saturated

Methane Tr = 1.1

Nitrogen Tr = 1.1

Figure 1: Concept of the law of corresponding state s The concept of the law of corresponding states is used in the development of compressibility charts and is the basis for many equations of state. Compressibility charts have been discussed in detail in Chemical Engineering 254 already and although not being discussed in this course, it should be noted that compressibility charts are a useful tool in describing PvT behaviour. Successful application of the principle of corresponding states to PvT data has encouraged similar correlations for other properties and a number of correlations given here will be based on the law of corresponding states.

1.2 Group contribution methods

Group contribution methods of estimating chemical properties originate from the concept that each chemical group in a compound contributes to the property in its own unique way. Although not without flaws, these methods are able to predict some properties with reasonable accuracy. State of the art group contribution methods, although quite complex, are accurate and reliable enough for most


engineering applications. A number of the simpler group contribution methods for the properties discussed will be illustrated here.

1.3 (Semi) Empirical correlations

A number of properties of compounds are fitted to (semi) empirical correlations. Sometimes these correlations have a theoretical basis with fitted constants (semi-empirical correlations) and sometimes the correlations are based on the best fit of a mathematical equation with little or no theoretical foundation (empirical correlations). Care should be taken when applying such correlations: The range of application is usually given as well as the units. When applied outside the range of application, incorrect / inaccurate results may be obtained. However, within the range of application, (semi) empirical correlations often provide a good method for estimating the properties of an equation.

2 PURE COMPONENT CONSTANTS

2.1 Critical constants (pressure, temperature, volu me, compressibility)

With regard to critical constants, a number of terms need to be defined as they are used extensively in predicting properties of substances. Critical point: The critical point is the point (pressure, temperature, specific volume and

compressibility) on the vapour pressure curve at which two co-existing phases become identical. This point is unique for every pure component.

Critical temperature(Tc): The critical temperature is the temperature at the critical point. This is the

temperature above which a gas can not be condensed by increasing the temperature, i.e. the temperature above which the liquid phase can not be formed irrespective of the pressure of the system. The critical temperature is important in determining the phase boundaries of compounds and is often a parameter in, amongst others, equations of state and in vapour pressure correlations.

Reduced temperature (Tr): The reduced temperature is the dimensionless ratio of the temperature to

the critical temperature (both in absolute values), i.e. Tr = T / Tc. Critical pressure (Pc): The critical pressure is the pressure at the critical point. This is the

pressure above which a gas can not be condensed by decreasing the temperature. As is the case for the critical temperature, the critical pressure is often used as parameter in equations of state and in vapour pressure correlations.

Reduced pressure (Pr): The reduced pressure is the dimensionless ratio of the absolute pressure

to the critical pressure, i.e. Pr = P / Pc. Critical volume (vc): The critical volume is the specific volume at the critical point, i.e. the

volume occupied by the compound at the critical temperature and pressure. The critical volume is sometimes used in equations of state as well as in estimating the volumetric fractions.

Critical compressibility (Zc) The critical compressibility is the compressibility at the critical point

calculated from the critical temperature, pressure and volume. The value is useful and often used in equations of state. (For methods estimating the critical compressibility please consult general references).

There are a number of literature sources for the critical constants. Some of them include (On reserve in the library):

• Perry’s Chemical Engineer’s Handbook, 7th Edition Chapter 2


• Properties of Gases and Liquids 4th Edition, Appendix A • CRC Handbook of Physics and Chemistry, Chapter 6

Many chemical thermodynamics textbooks, including the one prescribed for this course, also list some critical parameters.

Critical temperature The critical temperature is usually estimated using the normal boiling temperature. Generally the ratio between the critical temperature and normal boiling temperature is between 1.5 and 1.8. Guldberg’s rule states: Guldberg Tc = 1.5 Tb (1) Example: Estimate the critical temperature of cyclohexane using Guldberg’s Rule

Tb = 434.3 K Therefore Tc = 1.5 * 434.3 K = 651 K The published experimental value is 625 K (Reid and Prausnitz). The error is thus ≈ 4 %.

Prudhomm found that the sum of the normal melting temperature and normal boiling temperature is approximately equal to the critical temperature and proposed the following rule: Prudhomm Tc = Tb + Tm (2) Example: Estimate the critical temperature of water and ammonia using Prudhomm’s Rule:

Water: Tb = 373.15 K, Tm = 273.15 K Tc = 373.15 + 273.15 = 646.3 K The published experimental value is 647.3 K (Koretsky). The error is thus less than 0.2 %. Ammonia: Tb = 239.8 K, Tm = 195.4 K Tc = 239.8 + 195.4 = 435.2 K The published experimental value is 405.5 K (Reid and Prausnitz). The error is therefore greater than 7 %.

The above methods are rough and, as seen, are often not accurate enough. More accurate additive or group contribution methods can be used. One such method is that of Lydersen. This method of estimating the critical temperature of hydrocarbon and non-hydrocarbon organic compounds uses a combination of the normal boiling temperature and the molecular structure to estimate the critical temperature: Lydersen

( )

∆−∆+

=∑∑

2TT

bc

567.0

TT (3)

∆T is the contribution of each functional group to the critical temperature. The values of ∆T for each group is given in Table 1 in section 9.1


Example: Estimate the critical temperature of cyclohexanol using Lydersen’s method

OH

Tb = 434.3 K

∑∆T = 5 * (-CH2- cyclic) + ( /\C− cyclic) + (-OH alcohol)

= 5 * 0.013 + 0.012 + 0.082 = 0.159 From equation 3:

[ ] K6200159.159.0567.0

3.434T

2c =−+

=

The published experimental value is 625 K (Reid and Prausnitz). The error in the value is thus ≈ 1 %

Constantinou and Gani proposed a group contribution method that takes into account both the structural groups (first order) as well as structural isomerisation (second order). As the second order contributions are quite complicated, only the first order contributions will be discussed here (For more accurate predictions, the second order contributions can be considered – consult the original article [1]). The critical temperature can be calculated as follows: Constantinou and Gani

( )∑= cic nTln128.181

T (4)

Where n is the number of functional groups (i) present and Tci the contribution of group (i) present. The values of Tci are given in Table 2 in section 9.2. Example: Estimate the critical temperature of cyclohexane using the method of Constantiou and Gani

There are 5 –CH2– groups, i.e. 5 * 3.4920 = 17.460 There is 1 /

\C− group, i.e. 4.0330 There is 1 –OH group, i.e. 9.7292 Therefore the right hand side of equation 4 is 31.2222. Solving Tc, the critical temperature is 623 K. The error less than 0.5 %.

Critical pressure The critical pressure of organics can be estimated according to the method of Lydersen: Lydersen:

( ) )Pa(34.0

MW101325P

2p

c

∑∆+= (5)

∆p it the contribution of the functional group to the critical pressure and MW is the molar weight in g/mol. The values of ∆p for each group is given in Table 1 in section 9.1.


Example: Estimate the critical pressure of fluorotrichloromethane according to the method of Lydersen

M = 137.38 g/mol

∑∆P = (Fluoro) + 3 * (Chloro) + (Quatro subst. Carbon)

= 0.224 + 3 * 0.320 + 0.210 = 1.394 From equation 5:

( )MPa 63.4

394.134.0

38.137101325P

2c =+

⋅=

The published experimental value is 4.41 MPa (Reid and Prausnitz). The error is thus ≈ 5 %

Constantinou and Gani proposed a method to estimate the critical pressure, similar to their method of estimating the critical temperature: Constantinou and Gani

∑ ⋅=−−

)bar(Pn100220.03075.1P

1ci

c

(6)

Pci is the contribution of the functional group (i) to the critical pressure and n is the number of groups (i) present. The values of Pci are given in Table 2 in section 9.2. Example: Estimate the critical pressure of n-butanol according to the method of Constantinou and Gani

There is 1 –OH group, i.e. 0.005148 There are 3 –CH2– groups, i.e. 3 * 0.010558 There is 1 –CH3– group, i.e. 0.019904 The right hand side of equation 6 is 0.056726. Solving for Pc the critical pressure is 41.9 bar. The published value is 44.2 bar (Reid and Prausnitz). The error is thus ≈ 5 %.

The method of Constantinou and Gani, as used, only takes first order interactions into account. Greater accuracy can be obtained if second order interactions are taken into account, yet, as previously mentioned, this is beyond the scope of this course.

Critical volume Methods to calculate the critical volume are similar to those calculating the critical pressure and temperature. Fedors proposed a method for pure organic compounds: Fedors

∆+= ∑ kmolm0266.0v

3vc (7)

∆v is the contribution of the functional group to the critical volume. The values if ∆v is given in Table 4 in section 9.1


Example: Estimate the critical volume of 2-butanol according to the method of Fedors

The molecular formula is C4H10O. Therefore:

∑∆v = 4 * (carbon atom) + 10 * (hydrogen atom) + (oxygen atom)

= 4 * 0.034426 + 10 * 0.009172 + 0.018000 = 0.2474 vc = 0.0266 + 0.2474 = 0.2740 m3/mol. The published experimental value is 0.2690 m3/mol (Perry). The error is thus ≈ 2 %.

Other methods, similar to those shown for the critical temperature and pressure also exist and have similar accuracy.

Critical properties of mixtures The critical properties of mixtures are often calculated as the molar average value of the individual values. For example, the critical temperature of a mixture can be calculated as follows: ∑ ⋅= )n(Tx)mixture(T cc (8)

Please note that x is the MOLE fraction and Tc(n) is the critical temperature of pure compound n. Example: Estimate the critical pressure and temperature of LPG

Composition (mol%): propane = 70 %, iso-butane = 20 %, n-butane = 10 % Pure component critical temperature: propane = 369.8 K, iso-butane = 408.2 K, n-butane = 425.2 K Tc (LPG) = 0.7 * 369.8 + 0.2 * 408.2 + 0.1 * 425.2 = 383.0 K Pure component critical pressure: propane = 42.5 bar, iso-butane = 36.5 bar, n-butane = 38.0 bar Pc (LPG) = 0.7 * 42.5 + 0.2 * 36.5 + 0.1 * 38.0 = 40.9 bar

2.2 Normal boiling point

The boiling point is defined as the temperature at which the vapour pressure of the liquid is equal to the pressure of one atmosphere on the liquid i.e. the temperature at which the vapour pressure is 101325 Pa. However, if two values of vapour pressure close to 1 atm are available, the normal boiling point can be interpolated or extrapolated. This is discussed in more detail in the section on vapour pressures. The normal boiling point is often published together with critical constants. Good sources for the normal boiling point include:

• Perry’s Chemical Engineer’s Handbook, 7th Ed. Chapter 2 • Properties of Gases and Liquids 4th Ed., Appendix A (p 568 et seq.) • CRC Handbook of Physics and Chemistry, Chapter 3 and Chapter 6

In addition, some suppliers of fine chemicals, such as Sigma-Aldrich, often give the melting points of their compounds in their catalogues. Caution should be exercised when using values from older sources since the temperatures were often reported at prevailing conditions (0.95 – 0.97 atm).


A number of correlations exist linking the critical temperature and the normal boiling point. Some of these relationships have been discussed in section 2.1 and should the critical temperature be available, the normal boiling point can be determined. The normal boiling point can also be determined from the vapour pressure curve by setting the pressure equal to 101325 Pa and solving the temperature. The vapour pressure curve is discussed in section 3. Steil and Thodos proposed a method to calculate the boiling points of normal hydrocarbons: Steil and Thodos

( ) )K(N0742.01

11631209T

85.0b⋅+

−= (9)

Where N is the number of carbon atoms in the compound. Example: Normal boiling point of decane (10 carbon atoms) according to the method of Steil and Thodos

( ) K446100742.01

11631209T

85.0b =⋅+

−=

The experimental value is 447.3 K (Reid and Prausnitz). The error is thus less than 0.5 %.

The accuracy of correlations such as those of Steil and Thodos are quite good but, as is the case with the correlation of Steil and Thodos, these types of correlations are often limited to a few compounds. A number of group contribution methods exist to predict the normal boiling point. Constantinou and Gani also proposed such a group contribution method, this method being very similar to the ones for estimating the critical temperature and the critical pressure. The normal boiling point can be estimated as follows: Constantinou and Gani

( )∑ ⋅= bib Tnln359.204

T (10)

Where Tbi is the contribution of the functional group (i) and n the number of groups (i) present. The values of Tbi are given in Table 2 in section 9.2. Example: Estimate the normal boiling point of octanoic acid according to the method of Constantinou and Gani

There 1 –CH3 groups, i.e. 0.8894 There is 6 –CH2– group, i.e. 6 * 0.9225 There is 1 COOH group, i.e. 5.8337 The right hand side of equation 10 is thus 12.2581. Solving for Tb in equation 10 the normal boiling point is 512 K. The published experimental value is 512 K !

2.3 Acentric factor

The acentric factor (ω) is a measure of the shape of a molecule, though it also measures its polarity. The acentric factor is often used as a parameter in equations of state, hence its importance in thermodynamics and chemical engineering. By definition, the acentric factor is calculated from the reduced pressure at a reduced temperature of 0.7:


( ) )ensionless(dim000.1Plog 7.0T

satr r

−−=ω = (11)

Critical pressures and temperatures are estimated according to the aforementioned methods. The vapour pressure can be predicted according to the methods set out below or obtained experimentally. For small, spherical molecules the acentric factor is essentially zero and it increases as branching, molecular weight and polarity increases. For mixtures, the acentric factor is usually taken as the simple molar average value of the compounds in the mixture:

∑=

ω=ωn

1iiix (12)

Example: Calculate the acentric factor of the LPG mixture given above

Composition (mol%): propane = 70 %, iso-butane = 20 %, n-butane = 10 % Pure component acentric factors: propane = 0.153, iso-butane = 0.183, n-butane = 0.199 ω (LPG) = 0.7 * 0.153 + 0.2 * 0.183 + 0.1 * 0.199 = 0.164

The acentric factor is usually published together with the critical constants of a compound.

3 VAPOUR PRESSURE OF LIQUIDS

The vapour pressure at a set temperature is the pressure exerted by a pure component at equilibrium at this temperature when both liquid and vapour phases exist. The vapour pressure is a continuous function of temperature and the vapour pressure curve extends from the triple point to the critical point. Vapour pressures have been measured extensively, usually as part of thermophysical data of a compound. Perry’s Chemical Engineers’ Handbook contains a large amount of the thermophysical data. Another useful source of vapour pressure data is the NIST website. (See section 6). If the natural logarithm of the vapour pressure (ln(Psat)) is plotted as a function of the reciprocal of temperature (1/T) the relationship is often linear over a limited temperature range. This is shown in Figure 2.


-12

-10

-8

-6

-4

-2

0

2

4

6

0.000 0.004 0.008 0.012 0.016

1/T (1/K)

ln(P

sat)

(ln

(bar

)) Ethane

Hexane

Methane

Nitrogen

Water

Figure 2: Relationship between 1/T and ln(P sat) for ethane, hexane, methane, nitrogen and water (Data from NIST Website) If a linear relationship between ln(Psat) and 1/T is assumed, the vapour pressure curve can be expressed as follows:

( ) cT1

mPln sat += (13)

This is probably the simplest method for predicting the vapour pressure or for interpolating (and to a limited extent extrapolating) values. If two values on the vapour pressure curve are know, this method can be used. Referring to equation 13:

( ) ( )

21

sat2

sat1

T1

T1

PlnPlnm

−

−= (14)

( )1

sat1 T

1mPlnc −= (15)

Using equations 14 and 15, equation 13 can now be used to determine the vapour pressure.


Example: Estimate the vapour pressure of propane at 0 oC by interpolating between two values

Given: Tb = 231.03 K, Psat (300.00 K) = 9.9780 bar At normal boiling point (Conditions 1): ( )sat

1Pln = 11.526 and 1/T1 = 0.004328

At 300.1 K (Conditions 2): ( )sat2Pln = 13.813 and 1/T2 = 0.003333

2298003333.0004328.0813.13526.11

m −=−−=

474.21004328.0)2298(526.11c =⋅−−=

Therefore: ( ) 474.21T1

2298Pln sat +−= , T in K, P in Pa

At 0 oC, i.e. 273.15 K:

( ) 06.13474.2115.273

12298Pln sat =+−=

Psat = 469863 Pa = 4.70 bar. The NIST website gives the value as 4.75 bar. The error is thus less than 1.3 %

The Antoine equation is based on the same concept, yet due to the introduction of an additional parameter is more accurate: Antoine

( )CT

BAPln sat

++= (16)

A, B and C are regression constants and are unique to each compound. Table 5 in section 9.4 lists the parameters for some compounds. Example: Estimate the vapour pressure of carbon tetrachloride at 37 oC using the Antoine equation

For carbon tetrachloride with P in mmHg and T in K: A = 15.8742 B = -2808.19 C = -45.99 At 37 oC (308 K)

( ) 2375.599.45310

19.28088742.15Pln sat =

−−=

Therefore Psat = 188.2 mmHg ≡ 25.1 kPa The experimental value is 25.1 kPa !

However, the Antoine equation does not fit the data accurately much above the normal boiling point. More accurate regression equations are required. These functions usually express the natural logarithm of the vapour pressure as a function of temperature. Different sources use different equations. Reid and Prausnitz use three correlations to estimate vapour pressure data for more than 500 compounds. The vapour pressure of these compounds can be expressed by one of three equations: Reid and Prausnitz Equation 1

( )1xDxCxBxAx

PP

ln635.1

c

sat

−+++=

(17)

where

cT

T1x −= (18)


Reid and Prausnitz Equation 2

( ) ( )2

satsat

T

PDTlnC

TB

APln⋅+⋅+−= (19)

Reid and Prausnitz 3

( )CT

BAPln sat

+−= (20)

The values of A, B and C are given in Properties of Gases and Liquids by Reid and Prausnitz (On reserve in the Library). The correlations use T in K and Psat in bar. Equation 3 (equation 20) is in essence the Antoine equation. (See above for example). In equation 1 the pressure is calculated explicitly as a complex function of temperature. On the other hand, equation 2 needs to be solved iteratively. Examples of both equations 1 and 2 follow: Example: Plot the vapour pressure curve from 260 to 430 K for 2,2-Dimethyl-propane using the correlation in Reid and Prausnitz

Correlation 1: A = -6.89153, B = 1.25019, C = -2.28233, D = -4.74891 Tc = 433.8 K, Pc = 32.0 bar Using the above the vapour pressure was calculate and plotted: ln(Psat) is plotted as a function of 1/T and compared to data from the NIST website:

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0.002 0.0025 0.003 0.0035 0.0041/T with T in K

ln(P

sat)

with

Psa

t in

bar

Reid and Prausnitz

NIST data

As seen, the data generated with the correlation by Reid and Prausnitz agrees very well with the data on the NIST website


Example: Estimate the vapour pressure of trichlorofluoromethane at 400 K using the correlation published in Reid and Prausnitz

Correlation 2: A = 42.089, B = 4464.14, C = -4.753, D = 2138 (P in bar, T in K) Tc = 471.2 K, Pc = 44.1 bar For 400 K:

P guess LHS RHS Ratio Change20 2.9957 2.7185 1.1020 Increase Guess25 3.2189 2.7853 1.1557 Decrease Guess15 2.7081 2.6517 1.0213 Decrease Guess14 2.6391 2.6383 1.0003 Decrease Guess

13.9 2.6319 2.6370 0.9981 Increase Guess13.95 2.6355 2.6376 0.9992 Increase Guess13.98 2.6376 2.6380 0.9998 Increase Guess13.99 2.6383 2.6382 1.0001 Close enough

The NIST website gives the value as 14.074 bar. The accuracy is thus ≈ 0.6 %.

Perry published a similar correlation for over 230 compounds: Perry

( ) ( ) 5C43

21

sat TCTlnCT

CCPln ⋅+⋅++= (21)

The values of C1, C2, C3, C4 and C5 are given by Perry (page 2-50 – 2-54 in Table 2-6, 7th Edition). The correlations use T in K and P in Pa. Example: Plot the vapour pressure curve of hydrogen sulphide between 200 and 350 K using the correlation in Perry

Parameters: C1 = 85.584, C2 = -3839.9, C3 = -11.199, C4 = 0.018848, C5 = 1 Using the above the vapour pressure was calculate and plotted: ln(Psat) is plotted as a function of 1/T and compared to data from the NIST website:

10

11

12

13

14

15

16

0.0025 0.003 0.0035 0.004 0.0045 0.005

1/T (T in K)

ln(P

sat)

(P

sat

in P

a)

Perry

NIST

As seen, the data generated with the correlation by Perry agrees very well with the data on the NIST website

The above methods all estimate the vapour pressure by correlating experimental values and fitting them to a mathematical expression. Lee and Kesler proposed a method to estimate the vapour


pressure using only the critical temperature and pressure and the acentric factor of the compound as parameters.

( )( ) ( )( )1satr

0satr

satr PlnPlnPln ⋅ω+= (22)

Where

( )( ) 6rr

r

0satr T169347.0Tln28862.1

T09648.6

92714.5Pln ⋅+⋅−−= (23)

( )( ) 6rr

r

1satr T43577.0Tln4721.13

T6875.15

2518.15Pln ⋅+⋅−−= (24)

This method can be applied at reduced temperatures above 0.3 or above the melting point, whichever is higher and below the critical point and is most reliable between reduced temperatures of 0.5 and 0.95. Example: Estimate the vapour pressure of 1-butene at 98 oC

Pure component parameters: Tc = 146.4 oC, Pc = 4.02 MPa, ω = 0.1867

( )( ) 885.0

5.4191.371

1.2734.1461.27398

Tr ==+

+=

From equation 23:

( )( )

7227.0

885.0169347.0885.0ln28862.1885.0

09648.692714.5Pln 60sat

r

−=

⋅+⋅−−=

From equation 24:

( )( )

6190.0

885.043577.0885.0ln4721.13885.06875.15

2518.15Pln 61satr

−=

⋅+⋅−−=

From equation 22: ( ) 8383.06190.01867.07227.0Pln sat

r −=−⋅+−=

4325.0Psatr =

MPa 74.102.44325.0

PPP csatr

sat

=⋅=⋅=

Perry gives an experimental value of 1.72 MPa. The error is thus ≈ 1.2 %

4 THERMAL PROPERTIES

4.1 Heat capacity / enthalpy

Heat capacity is defined as the amount of energy required to change the temperature of a unit of mass or mole by one degree. Typical units include J/mol.K and J/kg.K. Heat capacity is usually given either at constant pressure (Cp) or at constant volume (Cv). Although heat capacity at constant volume is sometimes used, most engineering applications use heat capacity at constant pressure and we will thus focus on heat capacity at constant pressure. In thermodynamics and in heat transfer the heat capacity is often used as a function of temperature in the definition of enthalpy: dh = Cp dT (25)


It is thus very useful if the heat capacity is expressed as a function of temperature. A number of literature sources publish correlations which can be used to determine the heat capacity. These sources include but are not limited to:

• Perry’s Chemical Engineers Handbook, 7th Edition, Chapter 2 • Properties of Gases and Liquids 4th Edition, Appendix A • Basic Principles and Calculations in Chemical Engineering, 6th Edition • JANNAF Tables

When using such correlations, one needs to take note of the phase that the correlation refers to as well as the temperature limits of the correlation. There are also a number of sources that publish values of the heat capacity and / or enthalpy at specific temperatures and pressures. The sources include but are not limited to:

• Steam Tables • Perry’s Chemical Engineers’ Handbook, 7th Edition, Chapter 2

This section focuses on energy changes without phase changes. For the energy associated with phase changes, please see section 4.2.

Vapour heat capacity at constant pressure Harrison and Seaton proposed a method, valid from 300 to 1500 K, to calculate the heat capacity of an ideal gas based in the type and number of atoms in the molecule: Harrison and Seaton

Other15P14B13Al12Si11Br10

I9Cl8F7S6N5O4H3C21igp

nananananana

nanananananananaa)mol/J(C

++++++

++++++++= (26)

Where a1 to a15 are constant parameters (See section 1.1 in Table 6) and ni are the number of atoms of type i. Example: Estimate the ideal gas heat capacity of acetone (C3H6O) at 600 K

igpC =-4.61 + 17.5 * 3 + 10.5 * 6 + 17.5 * 1 = 128 J/mol.K

Daubert et al. reported the experimental value as 121.8 J/mol.K. The error is thus ≈ 5.5 %

However, the method of Harrison and Seaton is limited to ideal gases and does not provide the heat capacity as an explicit function of temperature. Rihani and Doraiswamy suggested a group contribution method to determine the heat capacity of vapours as a function of temperature. The heat capacity has the following temperature functionality:

32p dTcTbTaC +++= (27)

Where the parameters a, b, c and d are calculated by a group contribution method: ana i∑ ⋅= (28)

bnb i∑ ⋅= (29)

∑ ⋅= icnc (30)

∑ ⋅= idnd (31)


The group contribution parameters of ai, bi, ci and di are given in Table 7 to Table 12 in section 1.1 for the unit cal/mol.K. Note the units and that the values are often multiplied by a power of 10 to give reasonable values in the table. Example: The vapour heat capacity of 3-methyl-1,2-butaniene as a function of temperature and at 300 K according to the method of Rihani and Doraiswamy

Molecular structure:

CH2 C C

CH3

CH3

There is one /

\3 CCCH == group and two –CH3 groups. Therefore a = 2 * 0.6087 + 2.6308 = 3.8483 b = 2 * 2.1433E-2 + 4.1658E-2 = 8.4524E-2 c = 2 * (-0.0852)E-4 + (-0.2845)E-4 = -0.4549E-4 d = 2 * 0.001135E-6 + 0.007277E-6 = 0.009547E-6 The temperature dependent heat capacity correlation is thus:

3824p T109547.0T104549.0T084524.08483.3C −− ⋅+⋅−+= (cal/mol.K)

Using the correlation above the heat capacity at 300 K is estimated to be 25.37 cal/mol.K. The literature value (Weast) is 25.3 cal/mol.K. The error is thus ≈ 0.3 %

Liquid heat capacity at constant pressure Shaw proposed a group contribution method to estimate the liquid heat capacity at constant pressure at 25 oC: Shaw

∑= pip nCC (32)

Where Cpi is the contribution of functional group (i) and n the number of groups (i) present. The values of Cpi are given in Table 13 in section 9.7. Example: Estimate the heat capacity of hexane and heptane at 298.15 K using the method of Shaw

Hexane: Two –CH3 groups, i.e. 8.80 * 2 = 17.60 Four –CH2– groups, i.e. 7.26 * 4 = 29.04 Cp = 17.60 + 29.04 = 46.6 cal/mol.K This NIST website gives the value as 46.379 cal/mol.K The error is thus ≈ 0.5 % Heptane: Two –CH3 groups, i.e. 8.80 * 2 = 17.60 Four –CH2– groups, i.e. 7.26 * 5 = 36.30 Cp = 17.60 + 36.30 = 53.9 cal/mol.K This NIST website gives the value as 53.657 cal/mol.K The error is thus ≈ 0.5 %

The method of Shaw, although simple and quite accurate is limited to 25 oC. Růžička and Domalski proposed a group contribution method that is able to express the heat capacity of liquids as a second order polynomial function of temperature. The method is applicable between the melting temperature


and the normal boiling temperature yet extrapolation of up to 80 K beyond the range has no considerable effect. Růžička and Domalski

2

p

100T

d100

Tba

R

C

++= (33)

Where the parameters a, b and d are calculated by a group contribution method: ana i∑ ⋅= (34)

bnb i∑ ⋅= (35)

∑ ⋅= idnd (36)

The temperature is in K and the unit of Cp then depends in the unit of R. The values of ai, bi and di for linear aliphatic and aromatics hydrocarbons are given in Table 14 in section 9.8. Parameters for halogen, nitrogen, oxygen and sulphur groups are given in Table 15 in section 9.8. The method can be extended to include second order interactions. This is beyond the scope of this course. (If required, please consult the original texts [2,3].) Example: Determine a temperature dependent correlation for the heat capacity of hexane and estimate the heat capacity at 298.15 K

There are two –CH3 groups: aCH3 = 3.8452 * 2 = 7.6904 bCH3 = -0.33997 * 2 = -0.67994 dCH3 = 0.19489 * 2 = 0.38979 There are four –CH2– groups: aCH2 = 2.7972 * 4 = 11.1888 bCH2 = -0.054967 * 4 = -0.219868 dCH2 = 0.10697 * 4 = 0.42788 Therefore for hexane:

2p

100T

81767.0100T

89981.0879.18R

C

+−=

K.molJ

100T

81767.0100T

89981.0879.18314.8C2

p

+−⋅=

At 298.15 K:

K.molJ1.195

10015.298

81767.0100

15.29889981.0879.18314.8C

2

p

=

+−⋅=

The NIST website gives a value of 194.0 J/mol.K The error is thus ≈ 0.5 %

Heat capacity of mixtures As a first approximation the heat capacity of mixtures can be approximated by calculating the molar average of the constituent compounds:

∑=

=n

1ipiip )T(Cx)T(C (37)


This method works similar to the above discussed group contribution methods yet does not take enthalpy of mixing into account. For accurate approximations, the enthalpy of mixing needs to be incorporated.

4.2 Enthalpy of phase changes

When a substance changes phase, the enthalpy of the substance changes. This section concerns itself with the energy associated with phase changes.

Enthalpy (heat) of vapourisation (liquid to vapour) The enthalpy of vapourisation (∆hvap) is defined as the difference in the enthalpies of a unit of mole or mass of a saturated vapour and a saturated liquid of a pure component. Basically, the enthalpy of vapourisation is the energy required to vapourise a unit of mole or mass at the said temperature. Similarly, the enthalpy of vapourisation is the amount of energy released when a unit mole or mass is condensed at the said temperature. Enthalpies of vapourisation are often published together with saturated enthalpies in sources such as the steam tables and Perry. Pitzer et al. proposed a method based in the corresponding states approach that only requires the critical temperature and acentric factor: Pitzer et al.

( ) ( ) 456.0r

354.0r

c

vapT195.10T108.7

RT

h−⋅ω⋅+−=

∆ (38)

Example: Estimate the enthalpy of vapourisation for propionaldehyde at 350 K using the method of Pitzer et al.

Tc = 504.4 therefore Tr = 0.6939, ω = 0.2559 From equation 38:

( ) ( ) 289.66993.012559.095.106939.0108.7RT

h 456.0354.0

c

vap =−⋅⋅+−=∆

Thus: ∆hvap = 6.289 * 8.314 * 504.4 = 2.64 E4 J/mol ≡ 26.4 J/kmol The experimental value is 26.85 (Perry) and the error is thus less than 2 %

Riedel proposed a method of estimating the enthalpy of vapourisation at the normal boiling point: Reidel

−

−

⋅⋅⋅=∆

c

b

c

c

bcb,vap

TT930.0

1325.101

Pln

T

TTR039.1h (39)

In equation 39, the critical pressure is in kPa and the critical temperature in K.


Example: Estimate the heat of vapourisation at the normal boiling point for ethyl acetate using the method of Reidel

Tc = 523.3 K, Tb = 350.2 K, Pc = 3880 kPa Therefore:

mol/kJ 28.32mol/J 4E 228.3

3.5232.350930.0

1325.101

3880ln

3.5232.350

3.523314.8039.1h b,vap

==

−

−

⋅⋅⋅=∆

The experimental value is 32.23 kJ/mol (Perry). The error is thus ≈ 0.16 %

Constantinou and Gani proposed a group contribution method to calculate the enthalpy of vapourisation at 298 K: Constantinou and Gani

∑ ⋅+=∆ viK298,vap hn892.6h (kJ/mol) (40)

Where n is the number of functional groups (i) present and hvi the contribution of groups (i) present. The values of hvi are given in 9.2 in Table 3. Example: Estimate the enthalpy of vapourisation of methanol at 298 K according to the method of Constantinou and Gani

There is one –CH3 group, i.e. 4.116 There is one –OH group, i.e. 24.529 Using equation 40:

537.35529.24116.4892.6h K298,vap =++=∆ kJ/mol

The NIST website gives the value as 37.466 kJ/mol. The error is thus ≈ 5 %

The enthalpy of vapourisation decreases with temperature and is zero at the critical point. If the value of the enthalpy of vapourisation is know at one point, the method of Watson can be used to describe the enthalpy of vapourisation at any other temperature: Watson

38.0

2,r

1,r1,vap2,vap T1

T1hh

−−

∆=∆ (41)

Example: Estimate the enthalpy of vapourisation for ethyl acetate at 450.0 K using the normal boiling point as reference and using the method of Watson

∆hvap,1 = 32.23 kJ/mol at T1 = 350.2, Tr,1 = 0.6692 Tr,2 = 450.0/523.3 = 0.8599 Therefore according to equation 41:

mol/kJ 25.236692.018599.01

23.32h38.0

2,vap =

−−=∆

The experimental value is 23.16 kJ/mol (Perry). The error is thus ≈ 0.4 %


Perry also published correlations for the heats of vapourisation for over 230 compounds. The heat of vapourisation is listed as a function of temperature: Perry

( ) ( )2r4r32 TCTCCr1f T1Ch ++−=∆ (42)

Where C1 through C4 are component specific parameters, listed in table 2-193 page 2-156 to 2-160 (7th Edition) Example: Estimate the enthalpy of vapourisation for ethyl acetate at 450 K using the method of Perry

For ethyl acetate: C1 = 4.933E7 C2 = 0.3847 C3 = C4 = 0 Tr = 450.0 / 523.3 = 0.8599

( ) kJ/mol 2.23kmol/J 7E32.28599.017E933.4h 3847.0f ==−=∆

The experimental value is 23.16 kJ/mol !

Enthalpy (heat) of fusion (liquid to solid) The enthalpy of fusion ∆hfus is defined as the difference of the enthalpies of a unit mole or mass of a solid and liquid at its melting temperature and one atmosphere of pressure. Basically, the enthalpy of fusion is the amount of energy required to melt a solid at its melting temperature and one atmosphere of pressure. Similarly, the enthalpy of fusion is the amount of energy released when a liquid solidifies at its melting temperature and one atmosphere of pressure. Perry list the enthalpy of fusion for a number of compounds. Correlations regarding the enthalpy of fusion are not as accurate as those for the enthalpy of vapourisation and generally rely on group contribution methods. These group contribution methods are complicated and beyond the scope of this course. (For additional reading see page 2-250 and the tables on pages 2-252 and 2-253 in Perry, 7th Edition).

4.3 Enthalpy (heat) of formation

The enthalpy of formation is the energy required to form the compound at a specific temperature (usually at 298 K) from its elements. The enthalpy of formation is often used in chemical reaction thermodynamics, hence its importance here. The enthalpy of formation of a number of compounds is listed in sources such as Physical Chemistry Textbooks and in Properties of Gases and Liquids by Reid and Prausnitz. However, on occasion the values are not available and need to be estimated. Constantinou and Gani proposed a group contribution method to calculate the enthalpy of formation at 298 K: Constantinou and Gani

∑ ⋅+=∆ fiK298,f hn835.10h (kJ/mol) (43)


Example: Estimate the standard (298 K) enthalpy of formation for ethanol using the method of Constantinou and Gani

There is one –CH3 group, i.e. -45.947 There is one –CH2– group, i.e -20.763 There is one –OH group, i.e. -181.422 Using equation 43:

30.237422.181763.20947.45835.10h K298,f −=−−−=∆ kJ/mol

Atkins gives the value as -235.10 kJ/mol. The error is thus ≈ 1 %. Note: This value is the standards energy of formation in the VAPOUR phase. If you want to calculate the standard energy of formation in the liquid phase, you need to take the heat of vaporisation at 298 K into account.

The heat of formation can also be estimated from the heats of combustion (if available). This method is quite useful, especially for organic compounds containing N, S, O etc. Example: Estimate the heat of formation for liquid ethanol using heat of combustion of ethanol

The formation reaction is as follows: 2 C (graphite) + 1/2 O2 (g) + 3 H2 (g) → C2H6O (liq) Consider the combustion reactions of C, H2, and CH3NO2: C(graphite) + O2 (g) → CO2 (g) ∆hc = - 393.5 kJ/mol R1 H2 (g) + 1/2 O2 (g) → H2O (liq) ∆hc = - 285.8 kJ/mol R2 C2H6O (liq) + 3 O2 (g) → 2 CO2 (g) + 3 H2O (liq) ∆hc = -1366.8 kJ/mol R3 Consider 2 * R1 + 3 * R2 – R3: Chemical reaction same as the formation reaction ∆hf = 2 * (-393.5) + 3 * (-285.8) – (-1366.8) = -277.6 kJ/mol The literature value is – 277.6 kJ/mol Remember, this is for LIQUID ethanol, because that is the way the reactions are set up!

Similarly, Hess’s Law together with information on other chemical reactions can be used to determine the heat of formation. This was discussed in Engineering Chemistry 123.

5 PHYSICAL PROPERTIES

Most of the properties discussed above concern themselves with the thermodynamic and critical properties of substances. These are the properties most commonly used in thermodynamics. Substances do, however, also have a number of physical properties which are important in engineering calculations, for example viscosity, thermal conductivity, diffusivity and surface tension. The amount of data available regarding these properties varies and often, in the absence of reliable experimental data, these properties need to be estimated. Estimation of these properties is beyond the scope of this course yet similar methods as those used for the above discussed properties are used. For further information consult sources such as Perry’s Chemical Engineers’ Handbook and Properties of Gases and Liquids.

6 WEB SOURCES

With the coming of the electronic age, a large amount of data is available electronically. However, due to the abundance of data and ease of publication, one needs to exercise caution with unknown sources. The American National Institute of Standards and Technology (NIST) has a website with very reliable data. Thermodynamic and physical data for a number of pure components is available with


excellent accuracy and can be used with confidence. The website is accessible through the university network at: http://webbook.nist.gov/chemistry/fluid/

7 NOMENCLATURE

a Coefficient of heat capacity equation an Parameter in the Harrison and Seaton equation with n = 1 through 15 A Constant in the Antoine equation / Constant in the Reid and Prausnitz

vapour pressure correlation b Coefficient of heat capacity equation B Constant in the Antoine equation / Constant in the Reid and Prausnitz

vapour pressure correlation c Intercept of the vapour pressure correlation / Coefficient of heat capacity

equation C Constant in the Antoine equation / Constant in the Reid and Prausnitz

vapour pressure correlation Cn Constants in vapour pressure / enthalpy of vapourisation correlation in

Perry with n = 1 through 5 Cp Heat capacity at constant pressure Cv Heat capacity at constant volume d Coefficient of heat capacity equation D Constant in the Reid and Prausnitz vapour pressure correlation h Enthalpy hvi Contribution of group i to the enthalpy of vapourisation in the method

according to Constantinou and Gani. hfi Contribution of group i to the standard enthalpy of fusion in the method

according to Constantinou and Gani. m Gradient of the vapour pressure correlation MW Molecular Mass n Number of groups (i) in a group contribution method ni The number of atoms of type i in a molecule (used in the Harrison and

Seaton equation) N Number of carbon atoms in a linear hydrocarbon R Gas constant P Pressure T Temperature v Specific volume x Temperature function used in the vapour pressure correlation Z Compressibility ω Acentric factor ∆ Group contribution parameter for critical constants according to the

method of Lydersen / Fedors ∆hc Standard (298K, 1 atm) enthalpy of combustion ∆hf Standard (298K, 1 atm) enthalpy of fusion ∆hvap Enthalpy of vapourisation Superscripts ig Refers to the property of an ideal gas sat Refers to the property at saturated conditions Subscripts 1 Refers to the property at condition 1 2 Refers to the property at condition 2 298K Refers to the property at 298 K b Refers to the property at the normal boiling point bi Refers to the contribution of the group i to the normal boiling point

(Constantinou and Gani method) c Refers to the property at the critical point


ci Refers to the contribution of the group i to the critical temperature/pressure (Constantinou and Gani’s method)

i Contribution of group i m Refers to the property at the melting point p Refers to the contribution of the group to the critical pressure (Lydersen

method) r Refers to the reduced property t Refers to the contribution of the group to the critical temperature

(Lydersen method) v Refers to the contribution of the group to the critical volume (Fedors

method)

8 REFERENCES

The following general references are good sources of physical properties and their estimation: [1] Perry, R.H.; Green, D.W. (Ed) (1997) Perry’s Chemical Engineering Handbook, 7th Edition

(International Edition), McGraw-Hill [2] Reid, R.C.; Prausnitz, J.M.; Poling, B.E. (1987) Properties of Gases and Liquids, 4th Edition,

McGraw-Hill [3] Lide, D.R. (Ed.) (1997) CRC Handbook of Chemistry and Physics, 77th Edition, CRC Press Other References [1] Constantinou, L.; Gani, R. (1994) New Group Contribution Method for Estimating Properties of

Pure Compounds, AIChE Journal 40 (10) 1697 – 1710. [2] Růžička, V.Jr.; Domalski, E.S.(1993) Estimation of the Heat Capacities of Organic Liquids as a

Function of Temperature Using Group Additivity. I. Hydrocarbon Compounds, Journal of Physical and Chemical Reference Data 22 (3) 597 – 618.

[3] Růžička, V.Jr.; Domalski, E.S.(1993) Estimation of the Heat Capacities of Organic Liquids as a

Function of Temperature Using Group Additivity. I. Compounds of Carbon, Hydrogen, Halogens, Nitrogen, Oxygen, and Sulphur, Journal of Physical and Chemical Reference Data 22 (3) 619 – 655.


9 APPENDICES

9.1 Group contribution parameters for the Lydersen’ s method

Table 1: Group contribution paramters for the Lyder sen method to calculate T c and P c

(From Perry’s Chemical Engineers’ Handbook, page 2-343, Table 2-385)


9.2 Group contribution parameters for the Constanti nou and Gani’s method

Table 2: First order group contribution parameters for the estimation of critical properties and normal boiling points for the method of Constantino u and Gani (From AIChE Journal, volume 40, no 10, 1994, pages 1699 and 1700, Table 1)


Table 2 continued

Table 3: First order group contribution parameters for the estimation of thermodynamic properties and normal boiling points for the method of Constantinou and Gani (From AIChE Journal, volume 40, no 10, 1994, page 1700, Table 2)


9.3 Group increments for the Fedors’ method

Table 4: Group contribution parameters for the Fedo rs method to calculate v c (From Perry’s Chemical Engineers’ Handbook, 7th Edition page 2-344, Table 2-386)

9.4 Constants for the Antoine equation

Table 5: Constants for the Antoine equation to esti mate the vapour pressure of some compounds (From Basic Principles and Calculations in Chemical Engineering, 6th Edition, page 669, Table G1) NOTE: P in mmHg and T in K


9.5 Parameters for method of Harrison and Seaton

Table 6: Parameters for the method of Harrison and Seaton to calculate ideal gas heat capacity (From Perry’s Chemical Engineers’s Handbook, page 2-348, Table 2-387)

Note: Results and parameters may be interpolated between temperatures


9.6 Group contribution parameters for Rihani and Do raiswamy’s method

Table 7: Group contribution parameters for aliphati c hydrocarbon groups for estimation of vapour heat capacities according to the method of R ihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1, 1965, page 18, Table 1)

Table 8: Group contribution parameters for aromatic hydrocarbon groups for estimation of vapour heat capacities according to the method of R ihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1, 1965, page 18, Table 2)


Table 9: Group contribution parameters for contribu tions due to ring formation for estimation of vapour heat capacities according to the method o f Rihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1, 1965, page 19, Table 3)

Table 10: Group contribution parameters for oxygen containing groups for estimation of vapour heat capacities according to the method of R ihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1 page 1965, Table 4)

Table 11: Group contribution parameters for nitroge n containing groups for estimation of vapour heat capacities according to the method of R ihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1, 1965, page 19, Table 5)


Table 12: Group contribution parameters for sulphur containing groups for estimation of vapour heat capacities according to the method of R ihani and Doraiswamy (From Industrial and Engineering Chemistry Fundamentals, volume 4 no 1, 1965, page 19, Table 6)

9.7 Group contribution parameters for Shaw’s method

Table 13: Group contribution parameters for estimat ion of liquid heat capacity at 25 oC according to the method of Shaw (From Journal of Chemical and Engineering Data, volume 14 no 4, page 464, Table 2)


9.8 Group contribution parameters for R ůžička and Domalski’s method

Table 14: Group contribution parameters for linear hydrocarbon groups for estimation of liquid heat capacities according to the method of R ůžička and Domalski (From Journal of Physical and Chemical Reference Data, volume 22 no 3, page 604, Table 5)


Table 15: Group contribution parameters for halogen , nitrogen, oxygen and sulphur containing groups for estimation of liquid heat capacities acc ording to the method of R ůžička and Domalski (From Journal of Physical and Chemical Reference Data, volume 22 no 3, page 632 and 633, Table 8)


Table 15 continued