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Law of Mass Action and Its Thermodynamic Derivation CHAPTER 6 Thermodynamics – II 293 Law of Mass Action and Its Thermodynamic Derivation According to the law of mass action, the rate of a chemical reaction is directly proportional to the product of the activities or simply the active masses of the reactants each term raised to its stoichiometric coefficients. To understand the law of mass action in mathematical language, consider a reaction in which two reactants A and B react to form the product C and D i.e. 푎퐴 + 푏퐵 → 푐퐶 + 푑퐷 (33) Then the law of mass action says the rate of the above conversion should be 푅푎푡푒 ∝ [퐴]푎[퐵]푏 (34) 푅푎푡푒 = 푘[퐴]푎[퐵]푏 (35) Where k is the constant of proportionality and is typically labeled as rate constant of the reaction. However, the actual rate of the reaction may or may not be equal to what is suggested by the “law of mass action” because the actual rate law may have powers raised to the active masses different from their stoichiometric coefficients. Mathematically, the actual rate law for the reaction given by equation (33) is 푅푎푡푒 ∝ [퐴]훼[퐵]훽 (36) 푅푎푡푒 = 푘[퐴]훼[퐵]훽 (37) Now comparing equation (35) and equation (37); the law of mass action and actual rate law will give same results when 푎 = 훼 and 푏 = 훽; whereas different results will be observed when 푎 ≠ 훼 and 푏 ≠ 훽. Modern Definition of the Law of Mass Action The law of mass action can be used to study the composition of a mixture in a reversible reaction under equilibrium conditions. To do so, consider a typical reversible reaction i.e. 푎퐴 + 푏퐵 ⇌ 푐퐶 + 푑퐷 (38) Now, from the law of mass action, we know that the rate of forward reaction (Rf) and rate backward reaction (Rb) will be 푎 푏 푅푓 = 푘푓[퐴] [퐵] (39) 푐 푑 푅푏 = 푘푏[퐶] [퐷] (40) Where 푘푓 and 푘푏 are the rate constants for the forward and backward reactions, respectively. After equilibrium is reached, we have 푅푓 = 푅푏 (41) Copyright © Mandeep Dalal 294 A Textbook of Physical Chemistry – Volume I 푎 푏 푐 푑 푘푓[퐴] [퐵] = 푘푏[퐶] [퐷] (42) or 푐 푑 푘푓 [퐶] [퐷] (43) = 푎 푏 푘푏 [퐴] [퐵] Since the 푘푓 and 푘푏 are also constant at equilibrium, the ratio of the two is also a constant and is typically labeled as K or the equilibrium constant. Therefore, equation (43) is modified as 푐 푑 푘푓 [퐶] [퐷] (44) 퐾 = = 푎 푏 푘푏 [퐴] [퐵] All this leads to the modern definition of “law of mass action” that the ratio of the multiplication of molar concentrations of products raised to the power of their stoichiometric coefficients to the multiplication of the molar concentrations of the reactants raised to the power of their stoichiometric coefficients is constant at constant temperature and is called as “equilibrium constant”. It is also worthy to mention that equation (44) is also known as the “law of chemical equilibrium”. Thermodynamic Derivation of the Law of Mass Action In order to derive the law of mass action thermodynamically, recall the general form of a typical reversible reaction under equilibrium conditions in which reactants and products are ideal gases i.e. 푎퐴 + 푏퐵 ⇌ 푐퐶 + 푑퐷 (45) Now, as we know that the total free energy of the reactant (퐺푅) can be formulated as 퐺푅 = 푎휇퐴 + 푏휇퐵 (46) Where 휇퐴 and 휇퐵 are the chemical potentials of reactant A and B, respectively. Similarly, the total free energy of the products (퐺푃) can also be formulated i.e. 퐺푃 = 푐휇퐶 + 푑휇퐷 (47) It is also important to mention that the temperature and pressure are kept constant. Moreover, the free energy change of the whole reaction can be obtained by subtracting equation (46) from equation (47) i.e. 훥퐺푟푒푎푐푡푖표푛 = 퐺푃 − 퐺푅 (48) 훥퐺푟푒푎푐푡푖표푛 = (푐휇퐶 + 푑휇퐷) − (푎휇퐴 + 푏휇퐵) (49) Recalling the fact that the free energy change at equilibrium is zero, equation (49) is reduced to (푐휇퐶 + 푑휇퐷) − (푎휇퐴 + 푏휇퐵) = 0 (51) Now recall the expression of the chemical potential of the ith species in gas phase i.e. Buy the complete book with TOC navigation, Copyright © high resolution images and Mandeep Dalal no watermark. CHAPTER 6 Thermodynamics – II 295 0 휇푖 = 휇푖 + 푅푇 ln 푝푖 (52) 0 Where 푝푖 and 휇푖 are the partial pressure and standard chemical potential of ith species, respectively. Now using equation (52) in equation (51), we get 0 0 0 0 [푐(휇퐶 + 푅푇 ln 푝퐶) + 푑(휇퐷 + 푅푇 ln 푝퐷)] − [푎(휇퐴 + 푅푇 ln 푝퐴) + 푏(휇퐵 + 푅푇 ln 푝퐵)] = 0 (53) or 0 0 0 0 푐휇퐶 + 푐푅푇 ln 푝퐶 + 푑휇퐷 + 푑푅푇 ln 푝퐷 − 푎휇퐴 − 푎푅푇 ln 푝퐴 − 푏휇퐵 − 푏푅푇 ln 푝퐵 = 0 (54) 0 푐 0 푑 0 푎 0 푏 푐휇퐶 + 푅푇 ln 푝퐶 + 푑휇퐷 + 푅푇 ln 푝퐷 − 푎휇퐴 − 푅푇 ln 푝퐴 − 푏휇퐵 − 푅푇 ln 푝퐵 = 0 (55) 푐 푑 푎 푏 0 0 0 0 푅푇 ln 푝퐶 + 푅푇 ln 푝퐷 − 푅푇 ln 푝퐴 − 푅푇 ln 푝퐵 = −푐휇퐶 − 푑휇퐷 + 푎휇퐴 + 푏휇퐵 or 푐 푑 푎 푏 0 0 0 0 푅푇 ln (푝퐶푝퐷) − 푅푇 ln (푝퐴 푝퐵) = −[푐휇퐶 + 푑휇퐷 − 푎휇퐴 − 푏휇퐵] (56) (푝푐 푝푑) (57) 퐶 퐷 [ 표 표] 푅푇 ln 푎 푏 = − 퐺푃 − 퐺푅 (푝퐴 푝퐵) 푐 푑 (푝퐶푝퐷) 표 (58) 푅푇 ln 푎 푏 = −훥퐺푟푒푎푐푡푖표푛 (푝퐴 푝퐵) 표 표 Where 훥퐺푟푒푎푐푡푖표푛 is the standard free energy change of the reaction can be simply abbreviated as 훥퐺 only. Therefore, the equation (58) can be rearranged as given below. 푐 푑 표 (푝퐶푝퐷) 훥퐺 (59) ln 푎 푏 = − (푝퐴 푝퐵) 푅푇 푐 푑 훥퐺표 푝퐶푝퐷 − (61) 푅푇 푎 푏 = 푒 푝퐴 푝퐵 Now because 훥퐺표 is a function of temperature only and R is a constant quantity, the right-hand side can be put equal to another constant, say ‘Kp’. 훥퐺표 − (62) 푒 푅푇 = 퐾푝 From equation (61) and equation (62), we have 푐 푑 푝퐶푝퐷 (63) 퐾푝 = 푎 푏 푝퐴 푝퐵 Which is again the modern statement of “law of mass action” but in terms of partial pressures. Buy the complete book with TOC navigation, Copyright © high resolution images and Mandeep Dalal no watermark. 296 A Textbook of Physical Chemistry – Volume I Other forms of equation (63) can also be written depending upon the reactants and products involved. If the chemical potentials of the reactants and products are in mole fractions (xi) i.e. 0 휇푖 = 휇푖 + 푅푇 ln 푥푖 (64) Then equation (63) takes the form 푐 푑 푥퐶 푥퐷 (65) 퐾푥 = 푎 푏 푥퐴 푥퐵 Similarly, If the chemical potentials of the reactants and products are in molar concentrations (ci) i.e. 0 휇푖 = 휇푖 + 푅푇 ln 푐푖 (66) Then equation (63) takes the form [퐶]푐[퐷]푑 (67) 퐾 = 푐 [퐴]푎[퐵]푏 Which is the popular form of “law of mass action”. Copyright © Mandeep Dalal LEGAL NOTICE This document is an excerpt from the book entitled “A Textbook of Physical Chemistry – Volume 1 by Mandeep Dalal”, and is the intellectual property of the Author/Publisher. The content of this document is protected by international copyright law and is valid only for the personal preview of the user who has originally downloaded it from the publisher’s website (www.dalalinstitute.com). 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