# weight densities, integral of interaction and the total of variation derivatives in case of spherical kontinualnyh adsorbents

General view of integral of interaction for a spherical surface

In a spherical surface

Using the equation (5.13) and having placed limits for a case of a spherical surface, we will gain:

0 0

Where distance from a volume device adsorbtiva d3r ' = r ' 2sinθdr'dθdφдо surfaces adsorbentaposle integrations we will gain

Expression for blanket integral of interaction:

Outside of a spherical surface / a conglobate body

To similar case (5.18) only limits for r'изменятся with (0; R) on (R; ∞).

Weight densities

In a spherical surface

Using formulas (2.6) and (5.18), we will gain settlement expression for weight densities:

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Let's give short deductions of all weight densities.

Having substituted expression for o "" (s), we will gain:

After integration we will gain definitive expression:

Having opened limits and transferring to functions Hevisajda, we will write down expression in

Following view:

Spending integration on s, we will gain definitive expression:

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Using expression for At ' 1 ’, we will gain:

As r ' 2 = s2 + r2 - 2rscos α, and s = erscosa, it is simple to find expression for a vector s:

Considering (5.36), we will write down two last weight densities for a spherics case:

Outside of a spherical surface

To similar case inside (5), only with change of limits for r'с (0; R) on (R; ∞).

Interaction integral

In sphere

Applying formulas (5.4) and (5.29), we will gain expression for interaction integral:

Let's converse expression to a view convenient for integration:

Integrating on s, we will gain definitively expressions for integral in a spherical surface:

Outside of sphere

To similar case in sphere (5.37), but with change of limits for r'с (0; R) on (R; ∞).

The total of variation derivatives

In sphere

Using (5.6) and (5.14), we will write down the following formula:

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Applying the gained expression, we can calculate integrals from variation derivatives.

Having opened integral on s, we will gain:

Similarly for two following expressions:

Having opened integral on s, we will write down:

Integration on s, we will find definitive expression:

Considering earlier found (5.36), we will gain:

Integrating on a variable s, we will write down last required expression:

In sphere

To similar case inside (5), only limits on r'изменятся with (0; R) on (R; ∞).