#"pK"_b( :"CH"_3^(-)) < "pK"_b( :"NH"_2^(-)) < "pK"_b( "H":^(-)) < "pK"_b( "HC"-="C":^(-))#
Well, the #"pK"_b# of the base is the #"pK"_a# of the conjugate acid. So, first, let's compare the #"pK"_a#s.
1) #"pK"_a = 25#, for #"HC"-="CH"#
2) #"pK"_a = 35#, for #"H"_2#
3) #"pK"_a = 36#, for #:"NH"_3#
4) #"pK"_a ~~ 48#, for #"CH"_4#
Assuming that the #"pK"_a#s are reasonably accurate between hydrogen and ammonia, let's find the #"pK"_b#s and compare. But first, let's rationalize this.
A stronger acid is a weaker base. The higher the #"pK"_a#, the lower the #"pK"_b#.
Thus, we expect that the lowest #"pK"_b# goes to #"CH"_3^(-)#, i.e. it is the strongest base (just like a low #"pK"_a# indicates a strong acid).
That should make sense, because really, when do you ever see methane deprotonated? Acetylene, however, tends to be deprotonated by, say, #"NaNH"_2#, or #"NaH"#, to react with alkyl halides and form a lengthened alkyne. That does in fact demonstrate the stronger basicity of #"NH"_2^(-)# or #"H"^(-)# relative to acetylide.
1) #bb("pK"_b) = 14 - "pK"_a = bb(-11)#
2) #bb("pK"_b) = 14 - "pK"_a = bb(-21)#
3) #bb("pK"_b) = 14 - "pK"_a = bb(-22)#
4) #bb("pK"_b) = 14 - "pK"_a = bb(-34)#
And checking these with a second method, we should get the same thing.
#10^(-"pK"_a) = K_a#
1) #K_a = 10^(-25)#
2) #K_a = 10^(-35)#
3) #K_a = 10^(-36)#
4) #K_a = 10^(-48)#
#K_w = K_aK_b => K_b = K_w/K_a#
1) #K_b = 10^(-14)/(10^(-25)) = 10^11#
2) #K_b = 10^(-14)/(10^(-35)) = 10^21#
3) #K_b = 10^(-14)/(10^(-36)) = 10^22#
4) #K_b = 10^(-14)/(10^(-48)) = 10^34#
Thus:
1) #bb("pK"_b) = -log(10^11) = bb(-11)#
2) #bb("pK"_b) = -log(10^21) = bb(-21)#
3) #bb("pK"_b) = -log(10^22) = bb(-22)#
4) #bb("pK"_b) = -log(10^34) = bb(-34)#
Same result. So yes, our qualitative prediction made sense. We knew that the #"pK"_b# of #"CH"_3^(-)# would be lowest, i.e. that it would be the strongest base.
Therefore, in increasing order of #"pK"_b#, i.e. decreasing basicity, it would be:
#color(blue)("pK"_b( :"CH"_3^(-)) < "pK"_b( :"NH"_2^(-)) < "pK"_b( "H":^(-)) < "pK"_b( "HC"-="C":^(-)))#
