We are analyzing an electron in the second orbit of a hydrogen-like ion, He$^+$ (which is singly ionized helium). In this case, the nucleus has a charge of $Z = 2$, and the electron is in the $n = 2$ energy level.

To determine which of the given options is incorrect, let's calculate each property step by step.

1. Velocity of the electron

For a hydrogen-like atom, the velocity $v$ of an electron in orbit $n$ is given by: $v = \left(\frac{Z}{n}\right) \cdot v_0$ where $v_0 = 2.18 \times 10^8 \, \text{cm/s}$ (velocity of an electron in the ground state of hydrogen).

For He$^+$ ($Z = 2$) in the second orbit ($n = 2$): $v = \left(\frac{2}{2}\right) \cdot 2.18 \times 10^8 \, \text{cm/s} = 2.18 \times 10^8 \, \text{cm/s}$ This means that the velocity of the electron is $2.18 \times 10^8$ cm/s, not $2.18 \times 10^{10}$ cm/s. Therefore, Option A is incorrect.

2. Potential Energy (P.E.) of the electron

The potential energy of an electron in a hydrogen-like ion is given by: $\text{P.E.} = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2}$ For $Z = 2$ and $n = 2$: $\text{P.E.} = -\frac{(2)^2 \cdot 13.6}{2^2} = -\frac{4 \cdot 13.6}{4} = -13.6 \, \text{eV}$ So, the potential energy is $-13.6$ eV, not $-27.2$ eV. Therefore, Option B is incorrect.

3. Kinetic Energy (K.E.) of the electron

The kinetic energy of an electron in orbit is half of the magnitude of the potential energy: $\text{K.E.} = -\frac{1}{2} \cdot \text{P.E.}$ Since $\text{P.E.} = -13.6 \, \text{eV}$: $\text{K.E.} = -\frac{1}{2} \times (-13.6) = 13.6 \, \text{eV}$ Thus, the kinetic energy is $13.6$ eV, not $-13.6$ eV. Therefore, Option C is incorrect.

4. Radius of the orbit

The radius of an electron in the $n$-th orbit for a hydrogen-like ion is given by: $r_n = \frac{n^2 a_0}{Z}$ where $a_0 = 0.529 \, \text{Å}$ is the Bohr radius.

For $Z = 2$ and $n = 2$: $r_2 = \frac{(2)^2 \times 0.529}{2} = \frac{4 \times 0.529}{2} = 1.058 \, \text{Å}$ Therefore, the radius is $1.058$ Å, which matches the given value. Thus, Option D is correct.

Final Answer:

The incorrect options are:

• A, B, and C.

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Do you want further clarification on any of these steps or explanations?

Here are five related questions to explore:

1. How is the energy of an electron in a hydrogen-like atom calculated?
2. What is the relationship between potential energy and kinetic energy in atomic orbitals?
3. How does the atomic radius change with different atomic numbers in hydrogen-like ions?
4. Can the Bohr model be applied to multielectron atoms?
5. Why is the electron's kinetic energy always positive?

Tip: In hydrogen-like atoms, the potential energy is always twice the magnitude of the kinetic energy but negative, due to the attractive electrostatic force between the nucleus and the electron.