The value of the rate constant for the reaction between A & B was measured using pseudo-zeroth order conditions. The overall order of reaction was determined to be third order. When B was present at...?

#k_"obs" ~~ 5.56 xx 10^(-2) "M"^(-2)cdot"s"^(-1)#

The big picture is that the reaction is actually third order, with general rate law:

#r(t) = k_"obs"[A]^m[B]^n#, #" "m + n = 3#

where #k_"obs"# is the actual rate constant.

You're given that the order of #A# is #1#, so #B# must be second order. #m = 1#, so #3 - 1 = n = 2#.

Under conditions that are pseudo-zero order with respect to #B# (i.e. #[B]# #">>"# #[A]#), we have a rate constant of #k = 5.0 xx 10^(-3) "s"^(-1)# for the pseudo-first-order reaction where #A# is first order.

That is, we have:

#r(t) ~~ 5.0 xx 10^(-3) "s"^(-1) cdot [A] harr k_"obs"[A][B]^2#

Given what #[B]# is, and assuming that #[A]# is small compared to #[B]#, the actual rate law is supposed to be:

#r(t) = k_"obs"[A][B]^2#

That means the information about #[B]# is contained in #5.0 xx 10^(-3) "s"^(-1)#.

Under these same conditions, i.e. as long as #bb([B])# #bb">>"# #bb([A])# (usually #100+# times larger),

#k_"obs" cdot [B]^2 ~~ 5.0 xx 10^(-3) "s"^(-1)#

As a result, with #[B] = "0.30 M"# and #[A]# much smaller,

#color(blue)(k_"obs") ~~ (5.0 xx 10^(-3) "s"^(-1))/("0.30 M")^2#

#= color(blue)(5.56 xx 10^(-2) "M"^(-2) cdot "s"^(-1))#