Question

Consider an electron moving in the first Bohr orbit of a He\(^+\) ion with velocity \(v_1\). If it is allowed to move in the third Bohr orbit with velocity \(v_3\), determine the correct \(v_3 : v_1\) ratio.\\

• A. 3:1
• B. 2:1
• C. 1:3
• D. 1:2

Answer 1

The Correct Option is C

The velocity of an electron in a Bohr orbit is inversely proportional to the principal quantum number (n):

\( v \propto \frac{1}{n} \)

For the first Bohr orbit (n=1), the velocity is \(v_1\). For the third Bohr orbit (n=3), the velocity is \(v_3\). Using the proportionality:

\( \frac{v_3}{v_1} = \frac{1/3}{1/1} = \frac{1}{3} \)

Thus, the ratio \(v_3 : v_1\) is 1:3.

Answer 2

Option 3: 1:3

Explanation:

1. Bohr's Velocity Equation:

v = (Z * e²) / (2 * ε₀ * h * n)

where:

• v = velocity
• Z = atomic number
• e = elementary charge
• ε₀ = permittivity of free space
• h = Planck's constant
• n = principal quantum number

2. Velocity in the First Bohr Orbit (v₁):

For He⁺, Z = 2 and n = 1.

v₁ = (2 * e²) / (2 * ε₀ * h * 1) = e² / (ε₀ * h)

3. Velocity in the Third Bohr Orbit (v₃):

For He⁺, Z = 2 and n = 3.

v₃ = (2 * e²) / (2 * ε₀ * h * 3) = e² / (3 * ε₀ * h)

4. Determine the v₃:v₁ Ratio:

v₃ / v₁ = [e² / (3 * ε₀ * h)] / [e² / (ε₀ * h)] = 1/3

Therefore, v₃ : v₁ = 1 : 3