The Arrhenius Equation gives:

#sf(k=Ae^(-E_a/(RT)))#

#sf(k)# is the rate constant

#sf(A)# is the frequency factor, which is constant for a particular reaction

#sf(R)# is the gas constant, #sf(8.31"J/K/mol")#

#sf(T)# is the absolute temperature

Taking natural logs:

#sf(lnk=lnA-E_a/(RT))#

If #sf(T_1=800color(white)(x)K)# and #sf(T_2=700color(white)(x)K)# then:

#sf(lnk_1=lnA-E_a/(RT_1)" "color(red)((1)))#

and

#sf(lnk_2=lnA-E_a/(RT_2)" "color(red)((2))#

Subtracting both sides of #sf(color(red)((1)))# from #sf(color(red)((2))rArr#

#sf(ln[k_1/k_2]=-E_a/(RT_1)-(-E_a/(RT_2)))#

#:.##sf(ln[k_1/k_2]=E_a/R[1/T_2-1/T_1])#

Putting in the numbers:

#sf(ln[k_1/k_2]=(315xx10^3)/(8.31)[1/700-1/800])#

#sf(ln[k_1/k_2]=6.747)#

#:.##sf(k_1/k_2=851.728)#

#:.##sf(k_2=(9.7xx10^10)/851.728=1.1xx10^(8)color(white)(x)l.mol^(-1).s^(-1))#

This is less than #sf(k_1)# which is what you would expect since the reaction occurs at a lower temperature.