Final answer:
The energy emitted when an atom transitions from n=3 to n=1 is found using the formula E = -13.6 eV ((1/n²_1) - (1/n²_3)), resulting in the emission of 12.1 eV of energy for a hydrogen atom.
Step-by-step explanation:
Finding the Energy Emitted During an Electron Transition
To find the energy emitted when an atom transitions from n=3 to n=1, we use the following equation, which is derived from the Rydberg formula for hydrogen-like atoms:
E = -13.6 eV ((1/n²1) - (1/n²3))
Here, 13.6 eV is the ionization energy of a hydrogen atom when the electron is at the n=1 level, and n represents the principal quantum number of the electron's energy level. To calculate the emitted energy, substitute n1=1 and n3=3 into the equation:
E = -13.6 eV ((1/1²) - (1/3²))
= -13.6 eV (1 - 1/9)
= -13.6 eV (8/9)
= -12.1 eV
The negative sign indicates that energy is being released. Since we are interested in the amount of energy emitted, we can take the absolute value, which is 12.1 eV.
It's important to note that this calculation is specific to the hydrogen atom, and other atoms may have different ionization energies and would require different calculations.