51 eV of energy is supplied to He+ in ground state. Find out the total possible no. of spectral lines observed if transition takes place to ground state.

To determine the total number of spectral lines observed when 51 eV of energy is supplied to a helium ion (He⁺) in its ground state, we first need to understand the energy levels of the ion and how transitions between these levels lead to the emission of spectral lines.

Understanding Energy Levels in He⁺

Helium ion (He⁺) is a hydrogen-like atom, meaning it has only one electron. The energy levels of hydrogen-like atoms can be calculated using the formula:

E_n = -Z² * 13.6 eV / n²

Here, Z is the atomic number (for helium, Z = 2), and n is the principal quantum number corresponding to the energy level.

Calculating Energy Levels

For He⁺, the energy levels are:

• For n = 1: E₁ = -2² * 13.6 eV / 1² = -54.4 eV
• For n = 2: E₂ = -2² * 13.6 eV / 2² = -13.6 eV
• For n = 3: E₃ = -2² * 13.6 eV / 3² = -6.02 eV
• For n = 4: E₄ = -2² * 13.6 eV / 4² = -3.4 eV

And so on. The energy levels become less negative as n increases, indicating that the electron is less tightly bound to the nucleus.

Determining Possible Transitions

When 51 eV of energy is supplied to the He⁺ ion, the electron can be excited to higher energy levels. We need to find out how many energy levels can be reached with this energy.

Finding Excited States

The energy required to move from the ground state (n=1) to an excited state can be calculated as follows:

• From n=1 to n=2: ΔE = E₂ - E₁ = -13.6 eV - (-54.4 eV) = 40.8 eV
• From n=1 to n=3: ΔE = E₃ - E₁ = -6.02 eV - (-54.4 eV) = 48.38 eV
• From n=1 to n=4: ΔE = E₄ - E₁ = -3.4 eV - (-54.4 eV) = 51 eV

With 51 eV, the electron can be excited to n=4, as this is the maximum energy level it can reach with the supplied energy.

Calculating Spectral Lines from Transitions

Once the electron is excited to n=4, it can transition back to the ground state (n=1) through various paths. The possible transitions are:

• n=4 to n=1
• n=4 to n=2
• n=4 to n=3
• n=3 to n=1
• n=3 to n=2
• n=2 to n=1

Counting the Transitions

To find the total number of unique transitions, we can use the formula for combinations. The number of ways to choose 2 levels from n levels is given by:

C(n, 2) = n(n - 1) / 2

In this case, we have levels n=1, n=2, n=3, and n=4, which gives us 4 levels. Thus, the number of unique transitions is:

C(4, 2) = 4(4 - 1) / 2 = 6

Final Outcome

Therefore, the total number of spectral lines that can be observed when the electron transitions back to the ground state after being excited by 51 eV of energy is 6.