Atomic Spectra Test. 3

The kinetic energy of the electron (eV) is completely converted into the energy of the highest-energy photon (hf_max = hc/λ_min). So, eV = hc/λ_min. A useful shortcut is λ_min (in nm) ≈ 1240 / V (in volts). Thus, λ_min ≈ 1240 / 50000 = 0.0248 nm, which is approximately 0.025 nm.

Highest frequency means highest energy (E=hf). The largest possible energy drop in a hydrogen atom is from the ionization limit (n=∞) to the ground state (n=1).

In a three-level system, atoms are pumped to a high energy state, quickly decay to the metastable state, and then lase by transitioning from the metastable state back down to the ground state.

Rutherford's experiment, where alpha particles were deflected by a thin gold foil, led to the conclusion of a small, dense, positively charged nucleus.

The electron can cascade down in multiple steps. The total number of possible unique lines is given by the formula N = (n(n-1))/2, where n is the initial state. For n=4, N = (4(4-1))/2 = (4 × 3)/2 = 6. The possible transitions are 4→3, 4→2, 4→1, 3→2, 3→1, and 2→1.

The Paschen series ends at n=3. The 'first line' (longest wavelength) corresponds to the smallest energy jump, which is from the next level up, n=4.

Since each element has a unique spectral 'fingerprint', spectroscopy is a powerful tool for chemical analysis, like determining the composition of distant stars.

The longest wavelength (lowest energy) transition in the visible Balmer series is from n=3 to n=2. This specific line is called the H-alpha line and has a prominent red color.

Stimulated emission is the exact reverse: an incident photon causes an atom in an excited state to drop to the ground state, releasing a second photon.

A short wavelength implies high frequency and high photon energy (E=hf). This high energy allows them to pass through materials that block lower-energy visible light.

The mirrors trap photons, forcing them to pass through the active medium multiple times. This causes more stimulated emission and amplifies the light into a powerful beam.

This is derived from equating the kinetic energy of the electron (eV) to the maximum energy of the emitted photon (hf = hc/λ).

A ruby laser uses a powerful flash lamp (a light source) to excite the chromium ions in the ruby crystal. This method is known as optical pumping.

The energy levels are given by Eₙ = -E₀/n². As n gets larger, the energy levels get closer together, so the gap between them decreases.

X-rays are high-energy photons and are part of the electromagnetic spectrum, with wavelengths shorter than ultraviolet light but longer than gamma rays.

While Bohr's model successfully explained the spectra of single-electron systems like H and He⁺, it failed to account for several phenomena. These include the spectra of multi-electron atoms, the relative intensities of spectral lines, and the splitting of spectral lines in magnetic (Zeeman effect) and electric (Stark effect) fields.

In a three-level laser, the lower lasing level is the ground state, requiring over half the atoms to be pumped up. In a four-level laser, the lower lasing level is an intermediate excited state that is initially empty, making it much easier and less energy-intensive to create a population inversion.

Moseley's Law states that the frequency of characteristic X-rays is proportional to the square of the atomic number (√f ∝ Z). A higher Z means the inner-shell electrons are more tightly bound, leading to larger energy differences during transitions and thus higher frequency photons.

The radius of a Bohr orbit is directly proportional to the square of the principal quantum number (rₙ ∝ n²). Therefore, the ratio r₂ / r₄ = 2² / 4² = 4 / 16 = 1/4. The ratio is 1:4.

For a stable orbit, the electron must form a standing wave, which means the circumference (2πr) must contain an integer number of wavelengths (nλ). This condition, 2πr = nλ, directly leads to Bohr's postulate for the quantization of angular momentum.