01_-_Chemical_Kinetics_and_Equilibria

01_-_Chemical_Kinetics_and_Equilibria - Quantitative...

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Quantitative Physiology I / Molecular and Cellular Systems, BMEN E4001x Notes 01- Chemical Kinetics and Equilibria Law of Mass Action [ ] [ ] [ ] k * C * B * A reaction of rate D C C B A for 2 = → + + + k Note that A, B, C, etc, are thus in units of concentration, while k is of units to yield conc/time. Note that in the following, I’ll adopt K&S notation of lower case letters for species concentrations. Unimolecular reaction as a general framework – first order: B A k → A = reagent B = product k = rate constant, a mix of energetic and physical factors The reaction rate for this elementary reaction is defined as the rate of conversion of A to B, which is equivalent to either the rate of loss of A or rate of generation of B. In this simple case, ak k A dt da dt A d - = - = = ] [ ] [ From this, we get for a starting concentration of A of [A] 0 , or a 0 , ) kt exp( * ] A [ ) t ]( A [ 0 - = This is not a very useful chemical reaction in biology, as it is unregulated. This is found more often in decay processes. As a first level of complexity, add the reverse reaction B A 1 1 k k   → - This reaction is first order in both forward and reverse. An isomerization reaction, for example b k a k dt db b k a k dt da 1 1 1 1 - - - = + - = differential equations and 0 a b a = + “conservation” equation
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Solve, looking at the expression for a, and using the third relation 0 1 1 1 0 1 1 1 0 1 1 a ) k k ( k a ' a a k a ) k k ( ) a a ( k a k dt da - - - - - + - = + + - = - + - = which leads to 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 a ) k k ( k ) t ) k k ( exp( C ) t ( a ... or ) t ) k k ( exp( C ) t ( ' a .... or ' a ) k k ( a k a ) k k ( k ' a ) k k ( dt ' da - - - - - - - - - + + + - = + - = + - = + + + + - = For this example, make the assumption boundary condition that at t=0,a=a 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 a ) k k ( k C a k C ) k k ( a ) k k ( a ) k k ( k C a - - - - - - + = + + = + + + = Not the most informative, but let’s try to draw out. We know that a graph of [A] vs. t
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01_-_Chemical_Kinetics_and_Equilibria - Quantitative...

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