Final answer:
The total energy released when an electron transitions from n=3 to n=2 and then from n=2 to n=1 can be calculated using the formula En = -13.6 eV/n², yielding a total energy release of 12.09 eV.
Step-by-step explanation:
To calculate the energy when an electron falls from one energy level to another in a carbon atom, the energy of each level for carbon can be represented similarly to the hydrogen case (albeit incorrectly for the carbon's actual energy levels). Using the formula for hydrogen-like atoms, En = -13.6 eV/n², we can find the energy of the n=3 and n=2 levels. For n=3, the energy is E3 = -13.6 eV/3² = -1.51 eV, and for n=2, it's E2 = -13.6 eV/2² = -3.4 eV.
For the transition from n=3 to n=2, the energy released is ΔE = E2 - E3 = -3.4 eV - (-1.51 eV) = 1.89 eV. Similarly, for the transition from n=2 to n=1, the energy released is ΔE = E1 - E2, where E1 is the energy of the first level. Since E1 = -13.6 eV (the energy of the electron in n=1 for hydrogen-like atoms), the energy released for this transition is ΔE = -13.6 eV - (-3.4 eV) = 10.2 eV.
Adding the energy changes together: E_total = 1.89 eV + 10.2 eV = 12.09 eV. This is not an exact value for carbon, as carbon is not a hydrogen-like atom and this formula is based on hydrogen's spectral lines, but it gives a hypothetical scenario. There are differences in the energy levels due to the multielectron nature of carbon and its corresponding different shielding and electron-electron repulsions.
To address part (b), you would calculate the energy released in a direct fall from n=3 to n=1 for carbon, using the formula ΔE = E1 - E3. Finally, addressing part (c), the energies in part (a) and part (b) are not the same due to the difference in the initial and final states of the transitions.