Question

Energy levels \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a certain atom correspond to increasing values of energy, i.e., \(E_{\mathrm{A}}

Short answer

\( \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \)

Step-by-step solution

Understand the problem

We need to find the correct relationship between the wavelengths \(\lambda_1, \lambda_2\) and \(\lambda_3\) corresponding to the transitions between energy levels \(C \to B, B \to A\), and \(C \to A\), respectively, where the energy levels satisfy \(E_A < E_B < E_C\).

Recall the relationship between energy and wavelength

The energy of a photon emitted during a transition is given by \(E = \frac{hc}{\lambda}\). Therefore, the wavelength of the radiation is inversely proportional to the energy difference between the levels.

Write down the energy differences for each transition

For transition \(C \to B\): \(E_C - E_B = \frac{hc}{\lambda_1}\).For transition \(B \to A\): \(E_B - E_A = \frac{hc}{\lambda_2}\).For transition \(C \to A\): \(E_C - E_A = \frac{hc}{\lambda_3}\).

Combine the energy differences

Since \(E_C - E_A\) can be written as \( (E_C - E_B) + (E_B - E_A) \), and using the equations from the previous step, we get:\( \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \).

Simplify the expression

Dividing both sides by \(hc\) (which are constants), we have:\( \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \).Taking the reciprocal, we obtain:\( \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \).

Identify the correct option

Comparing with the given options, the correct statement is:Option (2) \( \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \).

Key concepts

Photon Energy

Photon energy is a fundamental concept when studying the behavior of light in atomic transitions. It's the energy carried by a photon, a particle representing a quantum of light. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.

The formula to find photon energy is given by:
\( E = h u \) or \( E = \frac{hc}{\lambda} \),
where:

• \(E\) is the photon energy
• \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{Js})\)
• \(u\) is the frequency of the photon
• \(\lambda\) is the wavelength of the light
• \(c\) is the speed of light in vacuum \((3.00 \times 10^8 \text{m/s})\)

Considering this formula, a smaller wavelength means higher photon energy because they are inversely related.

Understanding this relationship helps in solving problems involving atomic emission and transitions between different energy levels in atoms.

Wavelength Transitions

Wavelength transitions refer to the shifts in wavelength that occur when an electron moves between energy levels in an atom. When an electron drops from a higher to a lower energy level, it emits a photon whose wavelength corresponds to the energy difference between those levels.

Using the formula for photon energy, the wavelength of the emitted photon can be linked to the energy levels:\( \lambda = \frac{hc}{E} \).

In the context of the given problem, three transitions are discussed:

• \( \lambda_1 \): Transition from Level C to B
• \( \lambda_2 \): Transition from Level B to A
• \( \lambda_3 \): Transition from Level C to A

The relationship between these wavelengths comes from the atomic energy differences:\( \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \).
Upon manipulation, you get the expression:
\( \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \),
which is the correct relationship between the wavelengths for these transitions.

Atomic Emission

Atomic emission happens when an atom's electron transitions from a higher to a lower energy state, releasing energy in the form of light, or photons. This process explains why elements emit specific wavelengths of light, seen in their emission spectra.

When an electron shifts between discrete energy levels, it emits a photon with a particular energy, related to the difference between these energy levels.

There are key points to understand about atomic emission:

• Energy levels: Distinct states that electrons can occupy in an atom.
• Photon emission: Involves releasing a photon whose energy matches the gap between energy levels.
• Wavelength: The light emitted has a wavelength inversely related to the energy gap.

The emission spectrum of an element acts as its unique fingerprint because the wavelengths of emitted light depend on the differences between its energy levels. This concept is crucial for understanding phenomena like the hydrogen emission spectrum, and it forms the basis for techniques such as spectroscopy used in chemistry and astronomy.