The electron in hydrogen atom is excited to a higher energy level, where it stays for 10^-8 sec and makes 5.25×10^5 revolutions. Find the details Brogile wavelength of the electron in the same quantum level of He+ ion.

To find the de Broglie wavelength of the electron in the He⁺ ion, we first need to understand a few key concepts about atomic structure and wave-particle duality. The de Broglie wavelength is given by the formula λ = h/p, where h is Planck's constant and p is the momentum of the electron. For an electron in a hydrogen-like ion like He⁺, we can derive the momentum using its energy levels.

Understanding the Energy Levels

The energy levels of hydrogen-like ions can be calculated using the formula:

E_n = - (Z² * 13.6 eV) / n²

where:

• Z is the atomic number (for He⁺, Z = 2)
• n is the principal quantum number (the energy level of the electron)

For this calculation, we need to determine the energy level of the electron. Since the electron is initially in a higher energy state of hydrogen (where it makes 5.25 × 10⁵ revolutions in 10^-8 seconds), we can calculate the frequency (ν) of the electron's orbit:

Calculating the Frequency

Given that the electron makes 5.25 × 10⁵ revolutions in 10^-8 seconds, the frequency is:

ν = (5.25 × 10⁵ revolutions) / (10^-8 seconds) = 5.25 × 10¹³ Hz

Relating Frequency to Energy

The energy of the electron in a particular state can also be linked to its frequency using the formula:

E = hν

Here, h (Planck's constant) is approximately 6.63 × 10^-34 J·s. Thus, we can calculate the energy:

E = (6.63 × 10^-34 J·s) × (5.25 × 10¹³ Hz) ≈ 3.48 × 10^-20 J

Converting Energy to Electron Volts

To convert this energy to electron volts (1 eV = 1.6 × 10^-19 J), we find:

E ≈ 3.48 × 10^-20 J / (1.6 × 10^-19 J/eV) ≈ 0.2175 eV

Calculating Momentum

The momentum (p) of the electron can be derived from its energy using the relationship:

E = p² / (2m),

where m is the mass of the electron (approximately 9.11 × 10^-31 kg). Rearranging for p gives:

p = √(2mE)

Substituting the values:

p = √(2 × (9.11 × 10^-31 kg) × (3.48 × 10^-20 J)) ≈ 5.29 × 10^-24 kg·m/s

Finding the de Broglie Wavelength

Now we can find the de Broglie wavelength using the formula λ = h/p:

λ = (6.63 × 10^-34 J·s) / (5.29 × 10^-24 kg·m/s) ≈ 1.25 × 10^-10 m

Final Result

Thus, the de Broglie wavelength of the electron in the same quantum level of the He⁺ ion is approximately:

λ ≈ 1.25 × 10^-10 meters

This wavelength is quite small, consistent with the behavior of particles at the atomic level. The calculations showcase the relationship between energy, momentum, and wave properties of electrons in atomic systems.