# Help with these kinetics questions about reaction order with respect to one reactant #A#??

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#1)# The decay of #A# proceeds in a way such that the half-life is directly proportional to #[A]# . What is the order with respect to #A# ?

#2)# If doubling the concentration of reactant #A# quadruples the rate, then what would the order with respect to #A# be?

#3)# If doubling the concentration of #A# leads to a reaction rate that is #1.41# times as fast, what is the order with respect to #A# ?

#4)# Two initial concentrations of #A# were tested to determine the half-life, but both half-lives were equal even though the two concentrations were different. What is the order with respect to #A# ?

#5)# If the reaction rate is constant, what is the order with respect to #A# ?

##### 1 Answer

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#color(white)(0, 2, 1/2, 1, 0)#

Well, consider the following half-lives:

#t_"1/2" = ([A]_0)/(2k)# (zero order)

#t_"1/2" = (ln2)/k# (first order)

#t_"1/2" = 1/(k[A]_0)# (second order)

#t_"1/2" = 3/(2k[A]_0^2)# (third order)In all of these, only the zero order half-life is directly proportional to the concentration of

#A# . Hence, the reaction iszero orderwith respect to#A# .

You can figure out a lot from the rate law...

#r(t) = k[A]^n# Knowing that doubling the concentration leads to quadrupling the rate, i.e.

#color(red)(2)[A] -> color(red)(4)r(t)# , we have that

#color(red)(4)cdotr(t) = k(color(red)(2)[A])^n = color(red)(2)^n cdot k[A]^n# Thus, we have that

#2^n = 4# .What must#bbn# be?The reaction order with respect to#A# is ofthis#n# th order.

Same process as

#(2)# . Now we claim that:

#color(red)(1.41)cdotr(t) = k(color(red)(2)[A])^n = color(red)(2)^n cdot k[A]^n# But

#1.41 ~~ sqrt2# . Hence, we have that#sqrt2 = 2^n# .What is#bbn# this time?(What is#2# raised to that correlates with a square root?) The reaction is ofthis#n# th order with respect to#A# .

It discusses the first and second half-lives and claims that they are equal. This just says that

the half-life (of THIS order) does not depend on what concentration you start at.Look above. Which order half-life does not depend on

#[A]_0# ?Hint: it's not a prime number.

This says that the rate of reaction does not depend on the current

#[A]# . That is,#A# hasno influenceon the rate. That can only be the case if#A# is, i.e. the rate dependszero orderonlyon the rate constant:

#r(t) = k[A]^0 = k#