The charge/size ratio of a cation determines its polarizing power. Which one of the following sequences represents the increasing order of the polarizing power of the cationic species, $$K^+, Ca^{2+}, Mg^{2+}, Be^{2+}$$?

Polarising power is the ability of a cation to distort the electron cloud of an anion. It is directly proportional to the charge on the cation and inversely proportional to its ionic radius: $$\text{polarising power} \propto \frac{z^{+}}{r^{+}}$$, where $$z^{+}$$ is the positive charge and $$r^{+}$$ is the cationic radius.

Step 1 Compare charges.
$$K^{+}$$ has charge $$+1$$, while $$Ca^{2+},\; Mg^{2+},\; Be^{2+}$$ all have charge $$+2$$. Hence, with the same size, a doubly-charged ion would polarise more than a singly-charged ion.

Step 2 Compare sizes (radii decrease across a period and increase down a group).
Approximate ionic radii are
$$K^{+} \approx 138\text{ pm}$$, $$Ca^{2+} \approx 100\text{ pm}$$, $$Mg^{2+} \approx 72\text{ pm}$$, $$Be^{2+} \approx 31\text{ pm}$$.
Thus the size order is
$$K^{+} \gt Ca^{2+} \gt Mg^{2+} \gt Be^{2+}$$.

Step 3 Combine charge and size to get $$\frac{z^{+}}{r^{+}}$$ for each ion (qualitative comparison).
$$\frac{1}{138} \lt \frac{2}{100} \lt \frac{2}{72} \lt \frac{2}{31}$$.

Hence the polarising power increases in the sequence
$$K^{+} \lt Ca^{2+} \lt Mg^{2+} \lt Be^{2+}$$.

Therefore, the required increasing order is:
Option C which is: $$K^{+},\; Ca^{2+},\; Mg^{2+},\; Be^{2+}$$.