Question

How does the energy of a photon of visible light emitted by an atom compare to the energy change within the atom itself?

Short answer

The energy of a photon of visible light emitted by an atom is equal to the energy change within the atom itself. This is due to the conservation of energy principle, as the energy difference between the initial and final energy levels of an electron within the atom is released in the form of a photon. The energy of the emitted photon can be calculated using the formula \(E_{photon} = h \cdot \frac{c}{\lambda}\), where \(E_{photon}\) is directly related to the difference in energy levels in the atom: \(E_{photon} = E_{change} = E_i - E_f\).

Step-by-step solution

Understand conservation of energy within an atom

When an atom emits a photon of visible light, there is a corresponding energy change within the atom. This energy change occurs when an electron transitions from a higher energy level to a lower energy level, releasing the excess energy in the form of a photon. According to the conservation of energy principle, the energy of the photon must be equal to the difference in energy between the initial and final energy levels of the electron.

Identify the formula for the energy of a photon

The energy of a photon can be calculated using the formula: \(E = h \cdot f\) where: - \(E\) = energy of the photon - \(h\) = Planck's constant (\(6.63 \times 10^{-34}\) Js) - \(f\) = frequency of the emitted photon Alternatively, we can use the relationship between frequency and wavelength of the photon: \(f = \frac{c}{\lambda}\) where: - \(f\) = frequency of the emitted photon - \(c\) = speed of light (\(3 \times 10^{8}\) m/s) - \(\lambda\) = wavelength of the emitted photon Combining these two equations, we can calculate the energy of the photon using its wavelength: \(E = h \cdot \frac{c}{\lambda}\)

Equate the energy of the photon to the energy difference in the atom

Now that we have the formula for the energy of a photon, we can equate it to the energy difference between the initial and final energy levels of the electron in the atom. Let's denote the initial energy level as \(E_i\) and the final energy level as \(E_f\). The energy change within the atom can be represented as: \(E_{change} = E_i - E_f\) We can now equate the energy of the photon to the energy change within the atom: \(E_{photon} = E_{change}\) Substituting the formula for the energy of a photon: \(h \cdot \frac{c}{\lambda} = E_i - E_f\)

Compare the energy of the photon to the energy change within the atom

From the above equation, we can see that the energy of the photon of visible light emitted by an atom is equal to the energy change within the atom itself: \(E_{photon} = E_{change}\) This means that the energy of the emitted photon is directly related to the difference in energy between the initial and final energy levels of the electron within the atom. When an electron transitions from a higher energy level to a lower energy level, it releases a photon with an energy equal to the energy difference between these two energy levels.

Key concepts

Energy Transition

When an atom goes through energy transitions, it involves changes in the energy levels of its electrons. Imagine electrons as travelers on different floors of a building; each floor represents an energy level.

When an electron "jumps" from a higher floor (energy level) to a lower one, it emits energy in the form of light known as a photon. This is called an energy transition. This is similar to a basketball rolling down a hill, converting potential energy into kinetic energy.

The energy of the photon is directly related to the height of the jump, or the difference in energy levels of the atom. The bigger the jump, the more energy the photon has. Therefore, the photon carries away the exact amount of energy that the electron lost.

In simple terms, energy involved in transitions in atoms is the driving force behind the light that we see.

Conservation of Energy

The conservation of energy is a fundamental principle in physics. It states that energy cannot be created or destroyed in an isolated system. Imagine energy as money that can only be exchanged, never created out of thin air or made to vanish.

In the context of our atomic energy transition, this means that when an atom emits a photon, the total energy in the system remains the same. The energy lost by the electron during its jump gets converted entirely into the energy of the photon. This ensures that there is a perfect balance. The energy transition within the atom is essentially a rearrangement of energy

• The energy lost by the electron equals the energy gained by the photon.
• No energy is lost to the system overall; it is conserved.

In essence, the conservation of energy principle guarantees that the energy from the atom's transition is perfectly accounted for in the energy of the emitted photon.

Planck's Constant

Planck's constant is a fundamental number in physics that plays a crucial role in the study of quantum mechanics. It connects energy and frequency of photons, with a constant value denoted as \( h \), which is approximately \( 6.63 \times 10^{-34} \) Js.

Think of it as a scaling factor that enables us to calculate the energy of a photon if we know its frequency. The formula is straightforward:

• If you have the frequency \( f \), you can find the energy \( E \) using \( E = h \cdot f \).

Planck's constant ties together the wave and particle nature of light, essential for understanding phenomena at tiny scales.

For practical applications, such as calculating energy transitions in atoms, Planck's constant allows us to quantify the energy shifts during electron movements. It helps translate how much energy a photon will carry away during energy transitions.