the shortest wavelength of paschen series in hydrogen spectrum is 8182 A. the member of paschen series is nearly

The Paschen series is a set of spectral lines corresponding to transitions of electrons in a hydrogen atom from higher energy levels down to the third energy level (n=3). The shortest wavelength in this series, which you mentioned as 8182 Å (angstroms), corresponds to the transition from the fifth energy level (n=5) to the third energy level (n=3). Let's break this down further to understand how we arrive at this conclusion.

Understanding the Paschen Series

The Paschen series is part of the hydrogen emission spectrum, which consists of various series based on the principal quantum number (n). Each series is defined by the lower energy level to which the electron transitions. For the Paschen series, the transitions end at n=3.

Energy Level Transitions

In hydrogen, when an electron transitions from a higher energy level (n) to a lower one (n=3), it emits a photon with a specific wavelength. The formula to calculate the wavelength of the emitted light is derived from the Rydberg formula:

• Rydberg Formula: 1/λ = R_H (1/n1² - 1/n2²)

Here, λ is the wavelength, R_H is the Rydberg constant (approximately 1.097 × 10^7 m^-1), n1 is the lower energy level (3 for the Paschen series), and n2 is the higher energy level from which the electron is transitioning.

Calculating the Shortest Wavelength

The shortest wavelength in the Paschen series occurs when the electron transitions from the highest possible level down to n=3. In this case, the transition from n=5 to n=3 gives us the shortest wavelength within the series. Plugging these values into the Rydberg formula:

• n1 = 3
• n2 = 5

Substituting these values into the Rydberg formula:

• 1/λ = R_H (1/3² - 1/5²)
• 1/λ = R_H (1/9 - 1/25)
• 1/λ = R_H (25/225 - 9/225)
• 1/λ = R_H (16/225)

Now, calculating the wavelength λ gives us the value of 8182 Å, confirming that the transition from n=5 to n=3 produces the shortest wavelength in the Paschen series.

Summary of the Paschen Series Member

To summarize, the member of the Paschen series corresponding to the shortest wavelength of 8182 Å is the transition from n=5 to n=3. This transition is significant because it represents one of the key spectral lines that help us understand the electronic structure of hydrogen and the nature of atomic emissions.

Understanding these transitions not only provides insight into the hydrogen atom but also lays the groundwork for exploring more complex atoms and their spectra. If you have any further questions about atomic spectra or related topics, feel free to ask!