Question

What relation about elements did Dobereiner want to establish?

Answer 1

Dobereiner was the German chemist who worked on the periodic law of chemical elements and arranged the chemical elements in the form of triads. In his proposed triads, three elements were kept together as one family of elements. His work was considered a failure as it was not sufficient to explain more elements.

Complete answer:
According to Dobereiner’s theory, the atomic weight (or the atomic mass) of the central element is nearly or exactly equal to the mean of the atomic weights of the other two elements in that triad.
For example, in the case of Calcium (Ca - at. wt. $= 40.0$), Strontium (St at. wt. $= 87.6$), Barium (Ba at. wt. $= 137.3$)
The atomic weight of the central element,
$Sr = \left( {40 + 137.3} \right) = \left( {177.3} \right)/2 = 88.6$
As this calculated atomic weight of Strontium is near to its actual atomic weight, Dobereiner established that triads like Calcium, Strontium and Barium triad justifies the law.
But several other elements do not obey this law, hence, Dobereiner's triad system is considered a failure.

Note:
Dobereiner also discovered a pattern of the physical properties of the elements in the triad. He found that the atomic weight of the central element of each triad was approximately equal to the average atomic weight of the first and third elements. Dobereiner also found that the density of the central element of most triads is approximately equal to the average density of other elements.