How to Calculate Wavelength When N Is Increasing

Introduction

In quantum physics, the wavelength of an electron in an atom is determined by its energy state, which is described by the principal quantum number (n). As n increases, the electron moves to higher energy levels, which affects the wavelength of the electron's wave function.

The relationship between wavelength (λ) and the principal quantum number (n) is governed by the Rydberg formula, which describes the wavelengths of spectral lines of many chemical elements. This guide will explain how to calculate wavelength when n increases using the Rydberg formula.

The Formula

The Rydberg formula for the wavelength of light emitted or absorbed by an electron in a hydrogen atom is given by:

When calculating wavelength for increasing n, we're typically looking at transitions from lower to higher energy levels (n₁ < n₂).

Step-by-Step Calculation

1. Identify the initial and final principal quantum numbers (n₁ and n₂).
2. Square both quantum numbers (n₁² and n₂²).
3. Calculate the difference between the reciprocals of these squared numbers (1/n₁² - 1/n₂²).
4. Multiply this difference by the Rydberg constant (R∞).
5. Take the reciprocal of the result to get the wavelength in meters.

Worked Examples

Example 1: Transition from n=1 to n=2

Calculate the wavelength for a transition from the ground state (n₁=1) to the first excited state (n₂=2).

1. n₁ = 1, n₂ = 2
2. n₁² = 1, n₂² = 4
3. 1/n₁² - 1/n₂² = 1/1 - 1/4 = 0.75
4. 0.75 × R∞ = 0.75 × 1.0973731568539 × 10⁷ ≈ 8.23030367639925 × 10⁶ m⁻¹
5. λ = 1 / (8.23030367639925 × 10⁶) ≈ 1.21567 × 10⁻⁷ m or 121.567 nm

Example 2: Transition from n=2 to n=3

Calculate the wavelength for a transition from n=2 to n=3.

1. n₁ = 2, n₂ = 3
2. n₁² = 4, n₂² = 9
3. 1/n₁² - 1/n₂² = 1/4 - 1/9 ≈ 0.144444
4. 0.144444 × R∞ ≈ 0.144444 × 1.0973731568539 × 10⁷ ≈ 1.58198 × 10⁶ m⁻¹
5. λ = 1 / (1.58198 × 10⁶) ≈ 6.32 × 10⁻⁷ m or 632.4 nm