If the velocity of the electron in Bohr’s first orbit is 2.19 × 106 ms–1, calculate the de Broglie wavelength associated with it.

To find the de Broglie wavelength associated with an electron moving in Bohr's first orbit, we can use the de Broglie wavelength formula, which is given by:

De Broglie Wavelength Formula

The formula for calculating the de Broglie wavelength (λ) is:

λ = h / p

Where:

• λ = de Broglie wavelength
• h = Planck's constant (approximately 6.626 × 10^-34 Js)
• p = momentum of the particle

Calculating Momentum

The momentum (p) of the electron can be calculated using the formula:

p = m × v

Where:

• m = mass of the electron (approximately 9.11 × 10^-31 kg)
• v = velocity of the electron (given as 2.19 × 10⁶ m/s)

Now, let's calculate the momentum:

p = (9.11 × 10^-31 kg) × (2.19 × 10⁶ m/s)

Calculating this gives:

p ≈ 1.993 × 10^-24 kg m/s

Finding the De Broglie Wavelength

Now that we have the momentum, we can substitute it back into the de Broglie wavelength formula:

λ = h / p

Substituting the values:

λ = (6.626 × 10^-34 Js) / (1.993 × 10^-24 kg m/s)

Calculating this gives:

λ ≈ 3.32 × 10^-10 m

Final Result

Therefore, the de Broglie wavelength associated with the electron in Bohr's first orbit is approximately 3.32 × 10^-10 meters, or 0.332 nanometers. This wavelength is significant because it falls within the range of X-rays, illustrating the wave-particle duality of electrons.