Assume that the radius of the first Bohr orbit of hydrogen atom is $$0.6$$ $$\text{\AA}$$. The radius of the third Bohr orbit of He$$^+$$ is ______ picometer. (Nearest Integer)

We need to find the radius of the third Bohr orbit of $$\text{He}^+$$ given that the radius of the first Bohr orbit of hydrogen is $$0.6 \, \text{\AA}$$.

The Bohr radius formula is $$r_n = \dfrac{n^2}{Z} \cdot a_0$$.

In this formula, $$n$$ is the orbit number, $$Z$$ is the atomic number, and $$a_0$$ is the first Bohr radius of hydrogen.

For $$\text{He}^+$$, we have $$Z = 2$$, $$n = 3$$, and $$a_0 = 0.6 \, \text{\AA}$$.

Substituting these values into the formula gives $$r_3 = \dfrac{3^2}{2} \times 0.6 = \dfrac{9}{2} \times 0.6 = 4.5 \times 0.6 = 2.7 \, \text{\AA}$$.

Since $$1 \, \text{\AA} = 100 \, \text{pm}$$, we find $$r_3 = 2.7 \times 100 = 270 \, \text{pm}$$.

The radius of the third Bohr orbit of $$\text{He}^+$$ is $$\boxed{270}$$ picometers.