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The simple and basic answer is that they all react in the same way but more vigourously as you go down the column. Take the Group I elements, otherwise known as the Alkaline Metals. Firt in the column is Lithium. It reacts in water by whizzing around the surface until it disappears. Next is Sodium which melts and forms a ball and whizzes around the water until it disappears. Then Potassium will do the same as Sodium but will also burn with a flame as it whizzes around the water. Rubdinium will explode on contact with water as will Caesium but Caesium is more violent. Francium will probably be too violent a reaction to even attempt if enough could be gathered for this test.

The more involved answer would be is that within a group, the outer shell of electrons is always the same. As in this example above, Group I has 1 electron in its outer shell and it's always the electrons that determine the chemical behaviour of an atom.

The electron shells would be as follows:
Lithium 2,1
Sodium 2,8,1
Potassium 2,8,8,1
Rubdinium 2,8,8,18,1
Caesium 2,8,8,18,18,1
Francium 2,8,8,18,18,32,1

As the electrons want to be paired in the stated groups, then the outer 1 electron will always be lost more readily than 1 electron being gained (even though this can be done through Covalent bonding which allows electrons to move between atoms to satisfy the outer shell order).

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