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Integration of Heat Capacities

More ElaborateHeat Capacity Correlations for Wider Temperature

Ranges in 7

th

Edition of Perrys Chemical Engineering HandbookTable 2-198 p.2-178-182

Cp=C1+C2(C3/T/(sinh(C3/T))2+C4(C5/T/cosh(C5/T))

2

Temp in Degrees Kelvin

Cp Expressed in Joules/kmol/oK

sinh(C3/T) = (e(C3/T)

- e-(C3/T)

)/2

cosh(C5/T) = (e(C5/T)

+ e-(C5/T)

)/2

Refer to Table B.2 in Felder & Rousseau

(Form 1)CP= a +bT+CT2+dT

3

(Form 2) Cp= a + bT +C/T2

Cp(Kilojoules/Mol/oK)

Be Careful about Interpreting Constant Correctly

from Felder & Rousseau

Compound state Form Temp ax103bx10

5 cx10

8 dx10

12

Acetone l 1oC 123.0 18.6

g 1oC 71.96 20.10 -12.78 34.76

CaO c 2 K 41.84 2.03 -4.52x1010

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Cp(Acetone Liq) = 0.123+18.6x10-5

T where T inoC

Cp(Acetone Gas) = 0.07196+20.10x10-5

T-12.78x10-8

T2+34.76x10

-12 T

3

where T inoC

Cp(CaO Solid) = 0.04184+2.03x10-5

T +-4.52x102T

-2where T in

oK

Be careful with Cp (Liq Water)=75.4 x10-3

KJ/mol/oC=75.4KJ/Kmol/

oC

Other Heat Capacity Formulas

For Liquids & Solids (Cp & Cv) about the same

For Ideal Gases

Cp=Cv+R

Change in Specific Internal Energy

For Process Undergoing Change in Volume & Temperature

),(),( 2211 VTVT

dTCU

T

T

v2

1

Exact for Ideal GasesGood for Real Gases if Volume Constant

Good Approximation for Liquids & Solids

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Change in Specific Enthalpy when T & P both Change

(T1,P1) (T2,P2)

Consider Two Step Path

(T1,P1)(T1,P2)(T2,P2)For Ideal Gases

2

1

2

1

2

1

)()( 1

T

T

p

T

T

p

T

T

p dTCdTCRTdTCPVH

Also For Ideal & Real Gases if P Constant

2

1

T

T

pdTCH

However for Solids & Liquids

2

1

2

1

)(T

T

p

T

T

p dTCPVdTCVPH

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Use of Kopps Rule to Approximate Heat CapacitiesRefer to Perrys Handbook for Contributions of

Organic Functional Groups

Refer to Table B.10 (Atomic Heat Capacities)

Cpa (J/g-atomoC)

Element Solid Liquid

C 7.5 12

H 9.6 18

B 11 20

Si 16 24

O 17 25

F 21 29

P 23 31

S 26 31

All Others 26 33

Cp(SiO2Solid)= 16+2*17 = 50 J/Mol/oC

Cp(SiO2Liquid) = 24 + 2*25 = =74 J/Mol/oC

Enthalpy Changes of Mixtures of Gases or Liquids

80

25

80

25

)tan(75)(50 dTeLiqOcCdTDecaneLiqCH pp

Key Assumption: Ignoring the Heat of Mixing

Heater50 Moles/hr

C10H22(Liq)

75Moles

C8H18(Liq)

@25oC

50 Moles/hr

C10H22(Liq)

75Moles C8H18@80

oC

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For Very Dilute Solutions (Based Heat Capacity on Solvent &

Ignore Solute

Calculation of Change in Enthalpy

80

25

)(1000 dTDecaneLiqCH p

Key Assumption: Neglect Heat of Mixing

Calculation of Heat Capacity of a Mixture &

Enthalpy Change for Mixture(Neglect Heat of Mixing)

dTTCH

TCyTC

T

T

mixp

i

piimixp

2

1

)()(

)()()(

Heater1 MoleOctane(liq)

1000 Moles

Decane(liq)

@25oC

1 MoleOctane(liq)

1000 Moles

Decane(liq)

@80oC