The velocity of an electron in the first orbit of `H` atom is v. The velocity of an electron in the 2nd orbit of `He^+` atom.

Options

A. 2 v
B. v
C. v/2
D. v/4

Explanation

To find the velocity of an electron in the second orbit of the He⁺ atom, we can use the formula for the velocity of an electron in the nth orbit of a hydrogen-like atom:

\[ v_n = 2.19 \times 10^6 \frac{Z}{n} \text{ m/s} \]

where:
- \(v_n\) is the velocity of the electron in the nth orbit,
- \(Z\) is the atomic number of the atom,
- \(n\) is the principal quantum number (the orbit number).

### Step 1: Determine the velocity of the electron in the first orbit of the hydrogen atom (H).
For hydrogen (H):
- \(Z = 1\) (since hydrogen has one proton),
- \(n = 1\) (first orbit).

Using the formula:
\[ v_1 = 2.19 \times 10^6 \frac{1}{1} = 2.19 \times 10^6 \text{ m/s} \]

### Step 2: Relate this velocity to the variable \(v\).
From the problem, we know that the velocity of the electron in the first orbit of hydrogen is given as \(v\):
\[ v = 2.19 \times 10^6 \text{ m/s} \]

### Step 3: Calculate the velocity of the electron in the second orbit of the He⁺ atom.
For the helium ion (He⁺):
- \(Z = 2\) (since helium has two protons),
- \(n = 2\) (second orbit).

Using the formula:
\[ v_2 = 2.19 \times 10^6 \frac{2}{2} = 2.19 \times 10^6 \text{ m/s} \]

### Step 4: Relate this velocity back to \(v\).
Since we have already established that:
\[ v = 2.19 \times 10^6 \text{ m/s} \]
we can conclude:
\[ v_2 = v \]

### Final Answer:
The velocity of the electron in the second orbit of the He⁺ atom is \(v\).

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