Thermodynamics filled in class notes_Part_84

# Thermodynamics filled in class notes_Part_84 - 175 5.4...

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Unformatted text preview: 175 5.4. CHEMICAL EQUILIBRIUM Now writing the equilibrium condition in terms of the enthalpies and entropies referred to the standard pressure, one gets N i=1 o hT,i − T yi P Po so − R ln T,i N i=1 νi = 0, (5.258) N o hT,i − T so T,i νi = − i=1 =g o =µ o T,i T,i N − N g o νi T,i yi P Po RT νi ln = RT i=1 i=1 (5.259) νi yi P Po ln , , (5.260) ≡ ∆ Go − ∆Go = RT N i=1 N i=1 ∆Go exp − RT = i=1 , yi P Po , (5.262) νi yi P Po (5.261) νi yi P Po = ln N νi yi P Po ln νi , (5.263) , (5.264) ≡K P N KP = i=1 KP = P Po PN i=1 νi n ν yi i . (5.265) i=1 So N KP = i=1 Pi Po νi , at equilibrium. (5.266) Here KP is what is known as the pressure-based equilibrium constant. It is dimensionless. Despite its name, it is not a constant. It is deﬁned in terms of thermodynamic properties, and for the ideal gas is a function of T only: KP ≡ exp − ∆Go RT , generally valid. (5.267) Only at equilibrium does the property KP also equal the product of the partial pressures as in Eq. (5.266). The subscript P for pressure comes about because it is also related to CC BY-NC-ND. 18 November 2011, J. M. Powers. 176 CHAPTER 5. THERMOCHEMISTRY OF A SINGLE REACTION the product of the ratio of the partial pressure to the reference pressure raised to the net stoichiometric coeﬃcients. Also, the net change in Gibbs free energy of the reaction at the reference pressure, ∆Go , which is a function of T only, has been deﬁned as N o ∆G ≡ g o νi . T,i (5.268) i=1 The term ∆Go has units of kJ/kmole; it traditionally does not get an overbar. If ∆Go > 0, one has 0 < KP < 1, and reactants are favored over products. If ∆Go < 0, one gets KP > 1, and products are favored over reactants. One can also deduce that higher pressures P push the equilibrium in such a fashion that fewer moles are present, all else being equal. One can also deﬁne ∆Go in terms of the chemical aﬃnity, referred to the reference pressure, as ∆Go = −αo . (5.269) One can also deﬁne another convenient thermodynamic property, which for an ideal gas is a function of T alone, the equilibrium constant Kc : Kc ≡ PN Po RT i=1 νi exp − ∆Go RT , generally valid. (5.270) This property is dimensional, and the units depend on the stoichiometry of the reaction. PN The units of Kc will be (kmole/m3 ) i=1 νi . The equilibrium condition, Eq. (5.266), is often written in terms of molar concentrations and Kc . This can be achieved by the operations, valid only at an equilibrium state: N ρi RT Po KP = i=1 exp Po RT PN i=1 νi exp −∆Go RT −∆G RT = RT Po PN i=1 νi , (5.271) νi N ρi νi , (5.272) i=1 N o ρi νi . = (5.273) i=1 ≡K c So N ρi νi , Kc = at equilibrium. (5.274) i=1 One must be careful to distinguish between the general deﬁnition of Kc as given in Eq. (5.270), and the fact that at equilibrium it is driven to also have the value of product of molar species concentrations, raised to the appropriate stoichiometric power, as given in Eq. (5.274). CC BY-NC-ND. 18 November 2011, J. M. Powers. ...
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