Review. Benzene on the basis of the three-electron bond. Theory of three-electron bond in the four works with brief comments (review). 2016. - страница 7
E>2 = 3Ec—c +3Ec═c = 2890.286 kj/mole
The energy of six benzene c-c bonds with a 1.658 multiplicity is equal to:
E>3 = 6 · 534.0723 kj/mole = 3204.434 kj/mole
Therefore, the gain energy of benzene compared to cyclohexatriene will amount to:
E = E>3 – E>2 = 3204.434 kj/mole – 2890.286 kj/mole = 314.148 kj/mole (75.033 kcal/mole).
2.2. Experimental
Let’s show more detailed calculation of ratios for our mathematical relations. Let’s consider relation Multiplicity = f (L) and E = f (L) for С-С bonds, where multiplicity is multiplicity of bond, L – length of bond in Å, Е – energy of bond in kj/mole.
As initial points for the given bonds we will use ethane, ethene and acetylene. For the length of bonds let us take the findings [7]:
bond lengths in ethane, ethylene and acetylene
As usual, the С-С bond multiplicity in ethane, ethylene and acetylene is taken for 1, 2, 3. For the energy of bonds let us take the findings [7, p. 116]:
energies of bonds in ethane, ethylene and acetylene
If we have two variants and we received the set of points and we marked them on the plane in the rectangular system of coordinates and if the present points describe the line equation y = ax + b that for choose the coefficients a and b with the least medium-quadratic deflection from the experimental points, it is needed to calculate the coefficients a and b by the formulas:
(4)
(5)
n-the number of given values x or y.
If we want to know how big is the derivative, it is necessary to state the value of agreement between calculated and evaluated values y characterized by the quantity:
(6)
The proximity of r>2 to one means that our linear regression coordinates well with experimental points.
Let us find by the method of selection the function y = a + b/x + c/x>2 describing the dependence multiplicity = f (L) and E = f (L) in best way, in general this function describes this dependence for any chemical bonds.
Let us make some transformations for the function y = a + b/x + c/x>2, we accept
X = 1/x,
than we’ll receive: Y = b>1 + cX, that is the simple line equality, than
(7)
(8)
n—the number of given value Y.
Let us find a from the equality:
∑y = na + b∑ (1/x) + c∑ (1/x>2), (9)
when n = 3.
Let us find now multiplicity = f (L) for C─C, C═C, C≡C.
Table 1. Calculation of ratios for relation Multiplicity = f (L).