Thermodynamics filled in class notes_Part_86

Thermodynamics filled in class notes_Part_86 - 5.5....

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Unformatted text preview: 5.5. CHEMICAL KINETICS OF A SINGLE ISOTHERMAL REACTION 179 • The species concentrations of species involved in the reverse reaction decrease. Here, three intermediate variables which are in common usage have been defined. First one takes the reaction rate to be β r ≡ aT exp N −E RT ≡k (T ) ν′ ρk k k =1 forward reaction or β r = aT exp −E RT ≡k (T ), Arrhenius rate N 1 1 − Kc ν′ ρk k k =1 forward reaction N ρkνk k =1 reverse reaction 1 − Kc N ν ′′ ρk k k =1 reverse reaction law of mass action , . (5.280) (5.281) The reaction rate r has units of kmole/m3 /s. The temperature-dependency of the reaction rate is embodied in k (T ) is defined by what is known as an Arrhenius rate law: −E RT k (T ) ≡ aT β exp . (5.282) This equation was advocated by van’t Hoff in 1884; in 1889 Arrhenius gave a physical justification. The units of k (T ) actually depend on the reaction. This is a weakness of the theory, PN ′ and precludes a clean non-dimensionalization. The units must be (kmole/m3 )(1− k=1 νk ) /s. In terms of reaction progress, one can also take r= 1 dζ . V dt (5.283) The factor of 1/V is necessary because r has units of molar concentration per time and ζ has units of kmoles. The over-riding importance of the temperature sensitivity is illustrated as part of the next example. The remainder of the expression involving the products of the species concentrations is the defining characteristic of systems which obey the law of mass action. Though the history is complex, most attribute the law of mass action to Waage and Guldberg in 1864.1 Last, the overall molar production rate of species i, is often written as ωi , defined as ˙ ωi ≡ νi r. ˙ (5.284) 1 P. Waage and C. M. Guldberg, 1864, “Studies Concerning Affinity, Forhandlinger: Videnskabs-Selskabet i Christiania, 35. English translation: Journal of Chemical Education, 63(12): 1044-1047. CC BY-NC-ND. 18 November 2011, J. M. Powers. 180 CHAPTER 5. THERMOCHEMISTRY OF A SINGLE REACTION As νi is considered to be dimensionless, the units of ωi must be kmole/m3 /s. ˙ Example 5.13 Study the nitrogen dissociation problem considered in an earlier example, see p. 142, in which at t = 0 s, 1 kmole of N2 exists at P = 100 kP a and T = 6000 K . Take as before the reaction to be isothermal and isochoric. Consider again the elementary nitrogen dissociation reaction N2 + N2 ⇌ 2 N + N2 , (5.285) which has kinetic rate parameters of a = 7.0 × 1021 β = cm3 K 1.6 , mole s −1.6, E = (5.286) (5.287) cal 224928.4 . mole (5.288) In SI units, this becomes 3 a = E = m3 K 1.6 1m 1000 mole cm3 K 1.6 = 7.0 × 1018 , mole s 100 cm kmole kmole s cal J kJ kJ 1000 mole 224928.4 4.186 = 941550 . mole cal 1000 J kmole kmole 7.0 × 1021 (5.289) (5.290) At the initial state, the material is all N2 , so PN2 = P = 100 kP a. The ideal gas law then gives at t=0 P |t=0 ρN 2 t=0 = = PN2 |t=0 = ρN2 t=0 RT, P |t=0 , RT (5.292) 100 kP a , kJ 8.314 kmole K (6000 K ) kmole = 2.00465 × 10−3 . m3 Thus, the volume, constant for all time in the isochoric process, is = V= nN2 |t=0 1 kmole = ρN2 t=0 2.00465 × 10−3 (5.291) kmole m3 = 4.9884 × 102 m3 . (5.293) (5.294) (5.295) Now the stoichiometry of the reaction is such that − dnN2 = −(nN2 − nN2 |t=0 ) = 1 dnN , 2 1 (nN − nN |t=0 ), 2 ρN (5.297) =0 =1 kmole nN nN V (5.296) = = = = 2(1 kmole − nN2 ), 1 kmole nN2 2 − , V V 1 kmole − ρN 2 , 2 4.9884 × 102 m3 kmole − ρN 2 . 2 2.00465 × 10−3 m3 CC BY-NC-ND. 18 November 2011, J. M. Powers. (5.298) (5.299) (5.300) (5.301) ...
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Park-sou during the Fall '11 term at FSU.

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