Atomic physics is indeed a fascinating area of study, especially when you dive into the various equations that describe the behavior of atoms. Let's break down the key formulas you've mentioned, ensuring clarity and understanding as we go along. Each formula plays a significant role in explaining different phenomena in atomic physics.

De Broglie's Equation

The first formula you mentioned is De Broglie's equation, which expresses the wave-particle duality of matter:

λ = h / mv

Here, λ (lambda) represents the wavelength, h is Planck's constant (approximately 6.626 x 10^-34 Js), m is the mass of the particle, and v is its velocity. This equation suggests that every moving particle has an associated wavelength, which is a fundamental concept in quantum mechanics.

Planck's Equation

Next, we have Planck's equation, which relates energy to frequency:

E = hν or E = hc/λ

In this context, E is the energy of a photon, ν (nu) is the frequency, and c is the speed of light (approximately 3 x 10⁸ m/s). This equation is crucial for understanding how light interacts with matter, particularly in phenomena like the photoelectric effect.

Bohr's Model of the Atom

Bohr's model introduces several important conditions for electrons in atoms:

• mvr = nh/2: This condition describes the angular momentum of an electron in a stable orbit, where n is a positive integer (the principal quantum number).
• E₂ - E₁ = hν: This equation relates the energy difference between two levels to the frequency of the emitted or absorbed photon.

Radius and Energy of Orbits

For the radius of the nth orbit, we use:

r_n = n² * 0.53 x 10^-10 m

This formula indicates that the radius increases with the square of the principal quantum number. The energy of the nth orbit is given by:

E_n = -13.6/n² eV

This negative value indicates that the electron is bound to the nucleus, and the energy becomes less negative (increases) as n increases.

Frequency of Spectral Lines

The frequency of spectral lines can be calculated using the Rydberg formula:

ν = R [1/n₁² - 1/n₂²]

Here, R is the Rydberg constant (approximately 1.094 x 10⁷ m^-1), and n₁ and n₂ are the principal quantum numbers of the two energy levels involved in the transition. This formula is essential for predicting the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms.

Photoelectric Effect

Finally, the photoelectric effect is described by:

E = hν = W + (1/2)mv²

In this equation, W is the work function, which is the minimum energy required to remove an electron from the surface of a material. The threshold frequency (ν₀) corresponds to this work function.

Final Thoughts

Understanding these formulas is crucial for mastering atomic physics, especially for competitive exams like the JEE. Each equation not only serves a specific purpose but also connects to broader concepts in quantum mechanics and atomic theory. Keep practicing these formulas, and you'll find that they become second nature as you delve deeper into the subject!