Question

How does the energy possessed by an emitted photon compare to the difference in energy levels that gave rise to the emission of the photon?

Short answer

The energy of an emitted photon is equal to the difference in energy levels that gave rise to the emission of the photon. This can be expressed as \(E = E_2 - E_1\), where \(E\) is the energy of the photon, and \(E_2\) and \(E_1\) are the higher and lower energy levels, respectively.

Step-by-step solution

Understand the concept of energy levels

In an atom, electrons reside at various energy levels. When an electron transitions from a higher energy level (E₂) to a lower energy level (E₁), it emits a photon. The energy difference between these two energy levels is equal to the energy of the emitted photon.

Introduce Planck's formula

To determine the energy of an emitted photon, we use Planck's formula: \(E = h \cdot f\) where \(E\) is the energy of the photon, \(h\) is Planck's constant (approximately \(6.626 \times 10^{-34}\, Js\)), and \(f\) is the frequency of the emitted photon.

Relate frequency to energy difference

The frequency of the emitted photon (\(f\)) is related to the difference in energy levels (\(E_2 - E_1\)) by the formula: \(f = \dfrac{E_2 - E_1}{h}\)

Calculate the energy of the emitted photon

To calculate the energy of the emitted photon, we can substitute the expression for frequency found in Step 3 back into Planck's formula: \(E = h\cdot f = h \cdot \dfrac{E_2 - E_1}{h}\) The Planck's constants (h) will cancel out, and we are left with: \(E = E_2 - E_1\)

Conclusion

Based on the above derivation, we can conclude that the energy of an emitted photon is equal to the difference in energy levels that gave rise to the emission of the photon. In other words, for an electron transitioning from a higher energy level (E₂) to a lower energy level (E₁), the energy of the emitted photon will be equal to the difference between these two energy levels (\(E_2 - E_1\)).