**Calculating the Total Energy Evolved when all He+ ions return to the ground state**

Given information:
- 1 mole of He+ ions is excited.
- Spectral analysis showed 50% of the ions are in the 3rd level, 25% in the 2nd level, and the remaining 25% in the ground state.
- The ionization energy of He is 54.4 eV.

To calculate the total energy evolved when all the He+ ions return to the ground state, we need to consider the energy levels and their respective energies.

**Energy Levels and Energies:**
- Ground state (n = 1): E1
- 2nd level (n = 2): E2
- 3rd level (n = 3): E3

**Calculating the Energy Difference between Levels:**
The energy difference between levels can be calculated using the formula:

ΔE = |Ef - Ei|

Where ΔE is the energy difference, Ef is the final energy level, and Ei is the initial energy level.

- ΔE1-2 = |E2 - E1|
- ΔE2-3 = |E3 - E2|
- ΔE3-1 = |E1 - E3|

**Calculating the Energy Evolved for each Transition:**
The energy evolved for each transition is given by the equation:

Energy evolved = Percent in initial state * Number of ions * ΔE

- Energy evolved 1-2 = 25% * (1 mole) * ΔE1-2
- Energy evolved 2-3 = 50% * (1 mole) * ΔE2-3
- Energy evolved 3-1 = 25% * (1 mole) * ΔE3-1

**Calculating the Total Energy Evolved:**
The total energy evolved is the sum of the energies evolved for each transition.

Total energy evolved = Energy evolved 1-2 + Energy evolved 2-3 + Energy evolved 3-1

Substituting the calculated values in the equation, we can find the total energy evolved.

Remember to convert the energy difference from electron volts (eV) to joules (J) if necessary, using the conversion factor 1 eV = 1.6 x 10^-19 J.

Note: The specific values for the energy levels (E1, E2, and E3) are not provided in the question. You may need to refer to a periodic table or atomic data to obtain the accurate energy values.

Ans is 2× 10^25 eV.
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