Question

The structure of \(\mathrm{B_4H_{10}}\) is given below. Its styx number is:

• A. 4012
• B. 4102
• C. 2104
• D. 2014

Hint:

Count total H (10) minus one terminal H per boron (4) to get 6 extra H's split between bridges (s) and \(\mathrm{BH_2}\) groups (x); read the one B-B bond as y and check t=0 from the open butterfly skeleton.

Answer 1

The Correct Option is A

Step 1: Recall what styx counts.
Lipscomb's topological (styx) notation describes an electron-deficient borane using four numbers: \(s\) = number of 3-centre-2-electron \(\mathrm{B-H-B}\) bridge bonds, \(t\) = number of 3-centre-2-electron \(\mathrm{B-B-B}\) bonds, \(y\) = number of ordinary 2-centre-2-electron \(\mathrm{B-B}\) bonds, and \(x\) = number of \(\mathrm{BH_2}\) groups (borons carrying an extra terminal hydrogen beyond the one every boron already has).

Step 2: Work out \(p\) and \(q\) for \(\mathrm{B_4H_{10}}\).
Here \(p=4\) borons. Total hydrogens \(=10\), and since every boron already carries one terminal \(\mathrm{B-H}\) bond by default, the extra hydrogens are \(q=10-4=6\).

Step 3: Read the connectivity off the given structure.
The structure shows two central (bridgehead) boron atoms directly bonded to each other by a normal 2-centre bond, each carrying one terminal H. Each central boron is also linked to each of the two wingtip borons by a bridging hydrogen, giving 4 bridging \(\mathrm{B-H-B}\) bonds. Each wingtip boron carries two terminal hydrogens (a \(\mathrm{BH_2}\) group). This gives \(s=4\), \(t=0\) (no 3-centre \(\mathrm{B-B-B}\) bonds), \(y=1\) (the one direct \(\mathrm{B-B}\) bond between the bridgeheads), \(x=2\) (the two \(\mathrm{BH_2}\) wingtip borons).

Step 4: Check against Lipscomb's balance equations.
Extra-hydrogen balance: \(s+x=4+2=6\), matching \(q=6\). Boron-orbital balance, \(2s+3t+2y+x=3p\): \(2(4)+3(0)+2(1)+2=8+0+2+2=12=3(4)\), correct. Electron-pair balance, \(s+t+y+x=(2p+q)/2\): \(4+0+1+2=7\), and \((2\times4+6)/2=7\), correct.

Step 5: Write the styx number.
Writing the counts in the order \(s,t,y,x\) gives \(4,0,1,2\), i.e. 4012, option (A).

Final Answer:
The styx number of \(\mathrm{B_4H_{10}}\) is 4012. \[ \boxed{4012} \]