Find the binding energy of an electron in the ground state of a hydrogen like atom in whose spectrum the third Balmer line is equal to 108.5nm,

To find the binding energy of an electron in the ground state of a hydrogen-like atom, we can start by analyzing the information given about the third Balmer line. The Balmer series corresponds to transitions of electrons from higher energy levels to the second energy level (n=2) in hydrogen-like atoms. The third line in this series corresponds to a transition from n=5 to n=2.

Understanding the Balmer Series

The Balmer series is a set of spectral lines that result from electron transitions to the second energy level. The wavelengths of these lines can be calculated using the Rydberg formula:

1/λ = RZ²(1/2² - 1/n²)

Here, λ is the wavelength, R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹), Z is the atomic number, and n is the principal quantum number of the higher energy level from which the electron is transitioning.

Identifying the Transition

For the third Balmer line, the transition occurs from n=5 to n=2. Thus, we can plug in the values into the Rydberg formula:

1/λ = RZ²(1/2² - 1/5²)

Given that λ = 108.5 nm (which is 108.5 × 10⁻⁹ m), we can rearrange the formula to solve for Z:

1/λ = RZ²(1/4 - 1/25)

Calculating the Right Side

First, we need to compute the term inside the parentheses:

• 1/4 = 0.25
• 1/25 = 0.04
• 0.25 - 0.04 = 0.21

Now, substituting this back into the equation:

1/(108.5 × 10⁻⁹) = RZ²(0.21)

Solving for Z

Next, we can rearrange the equation to isolate Z:

Z² = (1/(108.5 × 10⁻⁹)) / (R × 0.21)

Substituting R = 1.097 × 10⁷ m⁻¹:

Z² = (1/(108.5 × 10⁻⁹)) / (1.097 × 10⁷ × 0.21)

Calculating the right side gives us:

Z² = (9.2 × 10⁶) / (2.3037 × 10⁶) ≈ 4.0

Taking the square root, we find Z ≈ 2. This indicates that we are dealing with a helium ion (He⁺), which is a hydrogen-like atom with one electron.

Finding the Binding Energy

The binding energy of an electron in a hydrogen-like atom can be calculated using the formula:

EB = -Z² × 13.6 eV

Substituting Z = 2:

EB = -2² × 13.6 eV = -4 × 13.6 eV = -54.4 eV

This negative value indicates that the electron is bound to the nucleus, and the magnitude represents the energy required to remove the electron from the atom.

Final Thoughts

In summary, the binding energy of the electron in the ground state of the hydrogen-like atom, which corresponds to the third Balmer line at 108.5 nm, is approximately 54.4 eV. This calculation illustrates the relationship between spectral lines and the energy levels of electrons in atoms, providing insight into atomic structure and behavior.