is their any way to find out to compare the chemical reactivity of two element while only using the periodic table

if you know the property trends than yes. But, be careful, because there are almost always exceptions to trends.

Just keep one thing in mind; Elements will always try and obtain a stable configuration of electrons. The closer an element is to having a full external shell of electrons, and the stronger it's attraction/repulsion to electrons are, the more reactive the element.

why does sodium (2,8,1) is less reactive then potassium(2,8,8,1) although both need 1 electron and also Na will more attract an electron due to small size

why does sodium (2,8,1) is less reactive then potassium(2,8,8,1) although both need 1 electron and also Na will more attract an electron due to small size

Actually, sodium and potassium both will more readily give up that s orbital electron, rather than gaining one...remember the stable octet "rule" (not really a rule, more of a guideline, as there can be cases where atoms have more than an octet of electrons)

why does sodium (2,8,1) is less reactive then potassium(2,8,8,1) although both need 1 electron and also Na will more attract an electron due to small size

 

Also keep in mind that as you increase the atomic number of the element, you increase the number of protons in the nucleus. All of the positive charge of an atom is located within the nucleus, while the negative charge is dispersed amongst the electrons outside of the nucleus. Since the positive charge is located inside the nucleus, an electron that is further away from the nucleus won't feel as much of a charge. It will be shielded from that pull by the negative charge of the electrons between it and the nucleus. (Positive always wants to be with negative).

 

While both K and Na have one electron in their outer shell, that outer electron in K has 8 more electrons between it and its nucleus. Even though K's nucleus has more positive charge, those 8 electrons do a good job of blocking that charge from that lone outer electron. As a result, that lone outer electron for K is easier to pull off than the one outer electron of Na. (Because that electron in Na doesn't have as much of a "shield" between it and the nucleus).

 

The final question I'll pose to you is what do you think is easier; grabbing 7 more electrons to get a full set of 8 outer electrons, or giving up just one to get a full outer shell?

Jdurg's way of reasoning is perfectly valid. For electron-gaining, the element fluorine is most reactive and going downwards along the halogens, the reactivity decreases. This can be explained, due to increasing radius of the atoms.

 

For the alkalimetals it is the other way around, but now electrons are given up. Cesium is large and the electron can more easily be given up, than by any other element and when going upwards, the giving up of the electron becomes less facile and hence the metal becomes less reactive.