Each boron is attached to at least one hydrogen atom by a two-electron sigma bond. The three-centered B-H-B bond is denoted by S, the closed or free three-centered B-B-B bond is expressed as t, the two-centered B-H_{t} and B-B bond is denoted by x and y, respectively.

As we know that boron compounds are electron-deficient compounds. But if each boron atom forms a three-centered bond, then the bond electron deficiency in boron hydride will be removed.

“styx” number helps us to understand the relation between the nature of bondings in borane and its structure.

Let, (BH)_{p}H_{q} is a general formula of a boron hydride. Here p = number of BH.

**Balance of three centers**:

The summation of three centered B-H-B bond and three centered B-B-B is equal to the number of BH units.

so, we can write

p = s + t ………………………………………………..(1)

**Balance of H atoms: **

Each boron atom is attached at least one terminal hydrogen atom. Therefore, remaining hydrogen atoms, q must be divided between bridging bonds(s), and additional B-H_{t} terminal bonds.

i.e q = s +x……………………………………………..(2)

**Balance of electrons:**

As we know, each bond consists of two electrons, so we can write:

number of electron pairs = number of bonds.

Each p(BH) group/unit gives two electrons whereas each additional hydrogen (q) atom gives only one electron.

For balancing the numbers of bonding electrons in borane, we can write the following:

p + q/2 = s +t+y+x ………………………………..(3)

From 1, 2, and 3 we get,

y = (s-x)/2……………………………………………..(4)

( The topology of boron hydrides/ boranes may be described in terms of the so-called “styx” number).

**Application of styx number: **

Find out the “styx” number of B_{2}H_{6} and draw its structure accordingly.

Step 1. Diborane can be represented as B_{2}H_{6}= (BH)_{2}H_{4} , so p = 2 and q = 4

Step 2. Here q = 4 and q/2 = 2, so that 2≤ s ≤ 4, The possible values of s = 2,3 and 4

Step 3. When p = 2, we get from equation (1) t = p-s = 2-2= 0 and also from equation 2, we get the value of x = q-s = 4-2 = 2

Step 4. After having f the value of x and s, we can calculate the value of y from equation 4, y = (s-x)/2 = (2-2)/2 = 0

So by considering the value of s= 2 , the first possible set of styx numbers is styx = 2002. Similarly, by taking value of s= 3 and 4 one will get two another set of possible styx numbers as described in the following table:

In order to select the correct the “styx” numbers among several possibilities, **empirical rules** have been developed which follow as:

- The “styx” number sets with negative value are not included as they have no physical significance.
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**2. Structure of B _{3}H_{9}**

Solution: B_{3}H_{9} can be expressed as (BH)_{3}H_{6} Hence, p = 3 and q = 6

Step 1: s = q/2 to q

= 3, 4, 5, and 6

Step 2: When s = 3, t = 0 y = 0 and x = 3

s = 4, t = -1, y = 1 and x = 2

s = 5, t = -2, y = 2, and x = 1

s = 6, t = -3, y = 3 and x = 0

From the above table, accepted styx number is 3003

S = 3 indicates 3 B-H-B bonds

t = 0 means no B-B-B bond

y = 0 represents no B-B bond

x = 3, indicates 3 -BH_{2} units

From the above information , the structure of B_{3}H_{9} will be:

STYX number of some boron hydrides have presented in the following table:

**3. Structure of B _{4}H_{10} **

Solution: B_{4}H_{10} can be expressed as (BH)_{4}H_{6} Hence, p = 4 and q = 6

Step 1: s = q/2 to q

= 3, 4, 5, and 6

Step 2: When s = 3, t= 1, y = 0, and x = 3 ( after calculation)

s = 4, t = 0, y = 1, and x = 2

s = 5, t = -1, y = 2, and x = 1

s = 6, t = -2, y = 3, and x = 0

The symmetric form of styx number 4012 is more stable than styx number 3103. So, Structure having styx number 4012 is accepted.

**4. Structure of B _{5}H_{9}**

Solution: B_{5}H_{9} can be expressed as (BH)_{5}H_{4} Hence, p = 5 and q = 4

Step 1: s = q/2 to q

= 2, 3 and 4

As we know, t = p-s, x = q-s and y =( s-x)/2

Step 2: When s = 2, we get (after calculation) t = 3, y = 0 and x = 2

s = 3, t = 2, y = 1 and x = 1

s = 4, t = 1, y = 2 and x =0

For s = 2 indicates 3 B-H-B bonds

t = 3 means 3 B-B-B bonds

y = 0 represents no B-B bond

x = 2 indicates 2 (BH2) units

For B_{5}H_{9} we get three valid sets of styx number. However, crystall data for B_{5}H_{9} compound supports the styx number 4120, which is more symmetric and more stable.

**5. Structure of B _{5}H_{11}**

Solution: Try yourself