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It is because of the size of the fluoride and chloride ion too. The fluoride ion is smaller than the chloride ion as size of an ion increases down a group. Therefore, the electrons are held more closely and hence, the bond is stronger.
It is mainly referring to the size of the atom. The larger the atom the larger the bonds, and the larger the bonds the weaker the bonds are. The smaller the atom the shorter the bonds, and the shorter the bonds the stronger the bonds are. So basically you just have to look at the atomic radius trend. The atomic radius decreases as you go left to right on the periodic table and increases as you go top to bottom. Fluorine is above Chlorine so it has a smaller atomic radius than Chlorine. Because it is smaller it will also have a shorter bond with another element. So Fluorine's bond with Carbon will be shorter and stronger than Chlorine's bond with Carbon.
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