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 possessed by an emitted photon, \(E_{photon}\), is equal to the difference in energy levels that gave rise to the emission of the photon, which can be represented as \(E_{photon} = E_{higher} - E_{lower}\). This relationship is derived from the principle of energy conservation and the formula for the energy of a photon \(E_{photon} = h * f\), where \(h\) is the Planck's constant and \(f\) is the frequency of the photon.

Step-by-step solution

Understand the given problem

First, we need to understand that an emitted photon is a result of an electron transitioning from a higher energy level to a lower energy level within an atom. The energy difference between these two energy levels usually corresponds to the energy of the emitted photon.

Energy conservation principle

Using the principle of energy conservation, we know that the energy cannot be created or destroyed but can only be transformed from one form to another. The energy lost by the electron as it moves to a lower energy state (level) must be equal to the energy gained by the photon, which is emitted as a result of this transition.

Formula for the energy of a photon

The energy of a photon can be calculated using the following formula: \(E_{photon} = h * f\), where: - \(E_{photon}\) is the energy of the photon, - \(h\) is the Planck's constant (\(6.626 \times 10^{-34} Js\)), - \(f\) is the frequency of the photon.

Relationship between the energy of an emitted photon and the energy difference of the energy levels

Following the conservation of energy principle, the energy of the emitted photon must be equal to the energy difference between the higher and lower energy levels of the electron due to its transition. Thus, we can write the relationship as: \(E_{photon} = E_{higher} - E_{lower}\), where: - \(E_{higher}\) is the energy level of the electron before transition, - \(E_{lower}\) is the energy level of the electron after transition.

Conclusion

The energy possessed by an emitted photon is equal to the difference in energy levels that gave rise to the emission of the photon. This follows directly from the principle of energy conservation and the formula for the energy of a photon.

Key concepts

Energy Levels

When discussing atoms, energy levels are the specific energies that electrons within the atom can have. Think of them as rungs on a ladder. Electrons can only exist in these defined energy levels; they cannot be in between. These levels are crucial because they determine how an electron behaves within an atom and during interactions such as photon emission.

Atoms can have multiple energy levels, and each level denotes a different state of energy for the electron. The further away an energy level is from the nucleus of the atom, the higher its energy. When an electron absorbs energy, it moves to a higher energy level, and when it loses energy, it falls back to a lower energy level. This behavior is foundational for understanding photon emission.

Electron Transition

In simple terms, electron transition refers to an electron moving between different energy levels within an atom. This movement is a result of energy gain or loss. When an electron absorbs energy, it jumps from a lower energy level to a higher one. Conversely, when an electron emits energy, it returns to a lower energy level.

Electron transitions are key for several physical phenomena, such as light emission. When an electron drops from a higher to a lower energy level, it emits a photon. The energy of this photon corresponds exactly to the energy difference between the two levels. Thus, each transition results in the emission of light with a specific frequency and wavelength, which is observable as different colors.

Energy Conservation

The principle of energy conservation is a fundamental concept in physics. It states that energy cannot be created or destroyed; it can only change from one form to another. In the context of photon emission, this principle explains that the total energy in a system remains constant.

When an electron transitions from a higher to a lower energy level, it releases energy. According to energy conservation, this loss must equal the energy gained by the emitted photon. Hence, the energy of the photon is directly linked to the energy difference between the electron's initial and final states. This conservation assures us that every bit of energy released by the electron is accounted for in the energy of the emitted photon.

Planck's Constant

Planck's Constant (\(h\)) is a fundamental quantity in quantum mechanics, which defines the energy scale in the quantum realm. It plays a pivotal role in the equation that calculates the energy of photons: \(E_{photon} = h \cdot f\). Here, \(h\) is Planck's Constant, while \(f\) is the frequency of the photon.

With a value of \(6.626 \times 10^{-34} Js\), Planck's Constant bridges the relationship between a photon's energy and its frequency, establishing a core connection between energy transitions and observable electromagnetic radiation. Its presence in calculations allows scientists to precisely determine how much energy is released or absorbed during such quantum processes. In essence, understanding Planck's Constant helps us make sense of how tiny particles like photons interact with matter in the universe.