Question regarding addition

Question

The beginning of the picture, from the word 'Acrylonitril' to the word 'structuurformule:', means the following: Acrylonitrle reacts with 1,4-butanediamine. When you use a lot of acrylonitril, under certain circumstances you almost exclusively get the following: Then you can see the first of the 2 structural formulas. The text between the 2 structural formulas, says This 'bond' can react with hydrogen. You'll get addition. When this addition finished, you get a bond with the following structural formula: And then you see the picture. The final blob of text: We can make polymere molecules which keep branching on and on because of reactions with hydrogen and acrylonitril. The final question is: * We start with 1 molecule of 1,4-butanediamine. Explain how many molecules of hydrogen are needed to make a polymere molecule with 16 $NH_2$ groups, assume that all reactions are complete.

The answer should be 56, but I have no clue why, can someone explain? I have half of 56, I just did 14x2, because you get 2 $H_2$ molecules per added Acrylonitril, and you have to get 14 Acrylonitril molecules to get to 16 $NH_2$ groups, so that would equal 2x14=28

Answer 1

This question appears to be about the preparation of polyacrylonitrile dendrimers. Each addition of acrylonitrile completely reacts away the NH₂ groups previously present:

assume that all reactions are complete.

Your answer of 28 assumes that the reactions could be controllable to allow partial consumption of NH₂ groups, which requires less addition of acrylonitrile and fewer equivalents of H₂. You need 16 NH₂ groups, which is 8 on each side. This is the third generation dendrimer:

• Generation 0 is butanediamine (2 NH₂ groups). Let $G_n$ represent the interior region of the $n^{th}$ generation. $$\ce{NH2CH2CH2CH2NH2} \equiv \ce{NH2}G_0\ce{NH2}$$
• Generation 1 involves reaction of 4 acrylonitriles followed by 8 molecules of hydrogen to make 4 NH₂ groups. $$\ce{NH2}G_0\ce{NH2 +4CH2=CHCN->(NCCH2CH2)2N}G_0\ce{N(CH2CH2CN)2} $$ $$\ce{(NCCH2CH2)2N}G_0\ce{N(CH2CH2CN)2 +8H2-> (NH2CH2CH2CH2)2N}G_0\ce{N(CH2CH2CH2NH2)2} \equiv \ce{(NH2)2}G_1\ce{(NH2)2} $$
• Generation 2 involves reaction of 8 acrylonitriles followed by 16 molecules of hydrogen to make 8 NH₂ groups. $$ \ce{(NH2)2}G_1\ce{(NH2)2 +8CH2=CHCN-> [(NCCH2CH2)2N]2}G_1\ce{[N(CH2CH2CN)2]2} $$ $$\ce{[(NCCH2CH2)2N]2}G_1\ce{[N(CH2CH2CN)2]2 +16H2 -> [(NH2CH2CH2CH2)2N]2}G_1\ce{[N(CH2CH2CH2NH2)2]2} \equiv \ce{(NH2)4}G_2\ce{(NH2)4}$$
• Generation 3 involves reaction of 16 acrylonitriles followed by 32 molecules of hydrogen to make 8 NH₂ groups. $$ \ce{(NH2)4}G_2\ce{(NH2)4 +16CH2=CHCN-> [(NCCH2CH2)2N]4}G_2\ce{[N(CH2CH2CN)2]4} $$ $$\ce{[(NCCH2CH2)2N]4}G_2\ce{[N(CH2CH2CN)2]4 +32H2 -> [(NH2CH2CH2CH2)2N]4}G_2\ce{[N(CH2CH2CH2NH2)2]4} \equiv \ce{(NH2)8}G_3\ce{(NH2)8}$$

Total number of H₂ molecules = 8 + 16 + 32 = 56 starting at $G_0$.