Question

An electron change its Position from orbit \(n=4\) to the orbit \(\mathrm{n}=2\) of an atom the wave length of emitted radiation in the form of \(\mathrm{R}\) (where \(\mathrm{R}\) is Redburg constants) (A) \((16 / 7 \mathrm{R})\) (B) \((16 / \mathrm{R})\) (C) (16 / 3R) (D) \((16 / 5 \mathrm{R})\)

Short answer

The wavelength of the emitted radiation when an electron changes its position from orbit n=4 to orbit n=2 is given by the formula \[\lambda = \frac{16}{3R}\]. Therefore, the correct answer is (C) 16 / 3R.

Step-by-step solution

Identify the given variables

In this problem, we are given the initial and final orbits of the electron: n₁ = 4 (Initial orbit) n₂ = 2 (Final orbit) The Rydberg constant for hydrogen is approximately R = 1.097 x 10^7 m⁻¹.

Plug in the values into the Rydberg formula

Now, we will plug in the values of n₁ and n₂ into the Rydberg formula: \[\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = R \left( \frac{1}{4^2} - \frac{1}{2^2} \right)\]

Simplify the equation

Now we will simplify the equation: \[\frac{1}{\lambda} = R \left( \frac{1}{16} - \frac{1}{4} \right)\] Next, find the common denominator (16) and subtract the fractions: \[\frac{1}{\lambda} = R \left( \frac{1-4}{16} \right)\] And finally, simplify the expression: \[\frac{1}{\lambda} = R \left( -\frac{3}{16} \right)\]

Solve for λ in terms of R

Now we need to solve for λ in terms of R. First, multiply both sides by -16/3: \[\lambda = -\frac{16}{3R}\] Since the wavelength cannot be negative, we can take the absolute value: \[\lambda = \frac{16}{3R}\] The wavelength of the emitted radiation is (C) 16 / 3R.

Key concepts

Understanding Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales, such as atomic and subatomic particles. It's a counterintuitive world where particles can exist in multiple states at once, and their properties aren't determined until they're measured. In quantum mechanics, probabilities and uncertainties play a key role. The concept of quantization is critical here—it indicates that certain physical properties (like energy) can only take on discrete values. This is unlike classical physics, where such properties can vary continuously. Key terms to keep in mind:

• Wave-particle duality: Particles can exhibit properties of both waves and particles.
• Uncertainty principle: You can't precisely know both the position and the momentum of a particle at the same time.
• Quantum state: The complete description of a quantum system.

Understanding these concepts helps us grasp how electrons transition between orbits and emit energy in the form of light.

Electron Orbits and Energy Levels

In quantum mechanics, electrons are not confined to fixed orbits like planets around the sun. Instead, they exist in regions called orbitals, where there is a high probability of finding an electron. These orbitals are defined by discrete energy levels, determined by quantum numbers. When an electron moves between orbits of different energy, it either absorbs or emits energy in the form of electromagnetic radiation. This concept is crucial for understanding spectral lines observed in emissions or absorptions spectra of atoms. Consider:

• Energies and orbits are quantized and depend on specific quantum numbers.
• An electron emits energy when dropping to a lower energy level and absorbs when moving to a higher level.
• These transitions result in electromagnetic radiation, which can be calculated using formulas like the Rydberg formula.

Such transitions explain why atoms emit light at specific wavelengths, and thus why we can use this light to infer atom properties.

Wavelength Calculation Using Rydberg Formula

The Rydberg formula is an important tool in quantum mechanics for calculating the wavelengths of light emitted or absorbed by electrons moving between energy levels in an atom. It is particularly useful for hydrogen and hydrogen-like atoms.The formula is given as:\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]Where:

• \( \lambda \) is the wavelength of the emitted or absorbed light.
• \( R \) is the Rydberg constant (for hydrogen, approximately 1.097 x 10^7 m⁻¹).
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final energy levels.

When an electron transitions from a higher to a lower orbit (e.g., from \( n=4 \) to \( n=2 \) in our example), the Rydberg formula helps compute the exact wavelength of light emitted. This wavelength corresponds to the energy difference between the two electron orbits, as derived from the change in their quantized energy levels.