Step 1: Understanding the Concept:
According to Bohr's model of the atom, electrons revolve around the nucleus in specific circular orbits. The radius of these orbits is quantized and depends on the principal quantum number ($n$).
Step 2: Key Formula or Approach:
The radius of the $n^{th}$ orbit for a hydrogen-like atom is given by:
\[ r_n = r_0 \times \frac{n^2}{Z} \]
where $r_0$ is the Bohr radius (radius of the first orbit of Hydrogen), $n$ is the principal quantum number, and $Z$ is the atomic number.
Step 3: Detailed Explanation:
1. For Hydrogen, $Z = 1$. The radius of the first orbit ($n=1$) is given as $r_1 = 0.530$ Å.
2. The "first excited state" corresponds to $n = 2$.
3. Using the proportionality $r_n \propto n^2$:
\[ r_2 = r_1 \times (2)^2 \]
\[ r_2 = 0.530 \times 4 \]
4. Calculation:
\[ r_2 = 2.120 \, \text{Å} \]
Step 4: Final Answer
The radius for the first excited state orbit is 2.12 Å.