Step-by-step Derivation:

1. Identify the Goal: The question asks for the kinetic energy (KE) of an electron in the second Bohr orbit ($n=2$) of a hydrogen atom, expressed in terms of Planck's constant ($h$), the electron's mass ($m$), and the Bohr radius (\\{\alpha_0}\\}).

2. Recall Key Formulas:

   • The kinetic energy of an electron is given by $KE = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity of the electron.
   • According to Bohr's model, the angular momentum of an electron in the n-th orbit is quantized: $L = mvr_n = \frac{nh}{2\pi}$, where $r_n$ is the radius of the n-th orbit.
   • The radius of the n-th Bohr orbit for a hydrogen atom ($Z=1$) is given by $r_n = n^2 \alpha_0$, where \\{\alpha_0}\\} is the Bohr radius (radius of the first orbit, $n=1$).

3. Express Velocity in terms of other variables: From the angular momentum quantization condition, we can solve for the velocity $v$: $v = \frac{nh}{2\pi m r_n}$

4. Substitute the velocity into the Kinetic Energy formula: Substitute the expression for $v$ into the KE formula: $KE = \frac{1}{2}m v^2 = \frac{1}{2}m \left(\frac{nh}{2\pi m r_n}\right)^2$ $KE = \frac{1}{2}m \left(\frac{n^2 h^2}{4\pi^2 m^2 r_n^2}\right)$ $KE = \frac{n^2 h^2}{8\pi^2 m r_n^2}$

5. Substitute the radius $r_n$ with its expression in terms of \\{\alpha_0}\\}: Now, substitute $r_n = n^2 \alpha_0$ into the expression for KE: $KE = \frac{n^2 h^2}{8\pi^2 m (n^2 \alpha_0)^2}$ $KE = \frac{n^2 h^2}{8\pi^2 m (n^4 \alpha_0^2)}$ Simplifying the expression by canceling $n^2$ from the numerator and denominator, we get a general formula for KE in the n-th orbit: $KE_n = \frac{h^2}{8\pi^2 m n^2 \alpha_0^2}$

6. Calculate the Kinetic Energy for the second Bohr orbit ($n=2$): The question specifically asks for the KE in the second Bohr orbit, so we set $n=2$: $KE_{n=2} = \frac{h^2}{8\pi^2 m (2)^2 \alpha_0^2}$ $KE_{n=2} = \frac{h^2}{8\pi^2 m (4) \alpha_0^2}$ $KE_{n=2} = \frac{h^2}{32\pi^2 m \alpha_0^2}$

7. Compare with the given options: The calculated value, ${h^2 \over {32{\pi ^2}m\alpha _0^2}}$, matches option C.

• A: {h^2 \over {4{\pi ^2}m\alpha _0^2}}}
• B: {h^2 \over {16{\pi ^2}m\alpha _0^2}}}
• C: {h^2 \over {32{\pi ^2}m\alpha _0^2}}}
• D: {h^2 \over {64{\pi ^2}m\alpha _0^2}}}

Thus, the correct option is C.