Chemical reactions obey a relationship between reaction rate k and activation en

Question

Chemical reactions obey a relationship between reaction ratek and activation energy G‡ that canbe written k =Qexp(-G‡/RT)where Q is a constant characteristic of the reaction,R is the gas constant, 0.001986 kcal/(mol-deg), andT is the Kelvin temperature. The activation energyG‡ for a particular uncatalyzedreaction is 15 kcal/mol and the overall difference in energybetween reactants and products G0 is0.

(a) By what factor will the reaction rate change if we increase thetemperature from 300K to 360K?
(b) An enzyme catalyzes the reaction so that the enzymatic reactionhas a G‡ = 5 kcal/mol. By whatfactor will the reaction rate increase at 300K if we compare thecatalyzed reaction to the uncatalyzed reaction?
(c) What is the physical significance of the constant Q inthe front of this equation? (hint: what will k be forT >> R/G‡)?

Explanation / Answer

a . Fomula :    ln(k2/k1) = (-Ea/R)[(1/T2)-(1/T1)]            Given data : Ea = 15kcal/mol =15x1000cal(4.2J/1cal)/mol   =63000J/mol                          T1= 300K                          T2= 360K                        ln(k2/k1) = (-Ea/R)[(1/T2)-(1/T1)]                                 = (-63000J.mol-1/8.314J.mol-1.K-1)[(1/360K)-(1/300K)]                                =   -7577.57K [-5.510-4]K-1                  ln(k2/k1) =41.75                    k2/k1 = e41.75                              = 1.35x1018             k2=   k1*1.35x1018   Thus , rateincreases by 1.35x1018 times . ..................................    b .  runcat = e(-Ea/RT)     and       rcat =e(-Ea-5kcal.mol-1/RT)                    rcat/runcat   =  e(-Ea-5kcal.mol-1/RT)/e(-Ea/RT)                             =  e(5kcal.mol-1/RT)                    rcat/runcat   =  e(-Ea-5kcal.mol-1/RT)/e(-Ea/RT)                             =  e(5kcal.mol-1/RT)                  Given data :   T =300K                                5kcal/mol= 5x1000cal(4.2J/1cal)/mol   =21000J/mol                                    rcat/runcat =  e(21000J.mol-1/8.314J.mol-1.K-1*300k)                                   = e2494.2                         rcat    =runcat e2494.2 .................................. c . At higher temperature ,k = Q                         rcat    =runcat e2494.2 .................................. c . At higher temperature ,k = Q

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