Find the longest and shortest wavelengths in the Lyman and Paschen series for hydrogen. In what region of the electromagnetic spectrum does each series lie?

Verified step by step guidance

1. Understand the concept of spectral series: The Lyman and Paschen series are sets of spectral lines corresponding to transitions of electrons in a hydrogen atom. The Lyman series involves transitions to the n=1 energy level, while the Paschen series involves transitions to the n=3 energy level.
2. Use the Rydberg formula to calculate wavelengths: The formula is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>&#x3BB;</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>R</mi><mi>H</mi></msub><mo>(</mo><mfrac><mn>1</mn><msup><mi>n</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mn>1</mn><msup><mi>m</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mfrac></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>H</mi></msub></math> is the Rydberg constant, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is the lower energy level, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the higher energy level.
3. Calculate the longest wavelength for each series: For the Lyman series, the longest wavelength occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> approaches infinity, resulting in a transition from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> = 1 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> = 2. For the Paschen series, the longest wavelength occurs for a transition from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> = 3 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> = 4.
4. Calculate the shortest wavelength for each series: For the Lyman series, the shortest wavelength occurs for a transition from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> = 1 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> = infinity. For the Paschen series, the shortest wavelength occurs for a transition from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> = 3 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> = infinity.
5. Identify the region of the electromagnetic spectrum: The Lyman series lies in the ultraviolet region, while the Paschen series lies in the infrared region. This is determined by the calculated wavelengths and their corresponding positions in the electromagnetic spectrum.

Key Concepts

Hydrogen Emission Spectrum

The hydrogen emission spectrum consists of several series of lines, each corresponding to electron transitions between energy levels. The Lyman series involves transitions to the n=1 level, while the Paschen series involves transitions to the n=3 level. These series are crucial for understanding the wavelengths emitted by hydrogen atoms.

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of spectral lines in hydrogen. It is given by 1/λ = R_H (1/n1^2 - 1/n2^2), where λ is the wavelength, R_H is the Rydberg constant, and n1 and n2 are the principal quantum numbers of the electron's initial and final states. This formula helps determine the longest and shortest wavelengths in the Lyman and Paschen series.

Electromagnetic Spectrum Regions

The electromagnetic spectrum is divided into regions based on wavelength, including radio, microwave, infrared, visible, ultraviolet, X-rays, and gamma rays. The Lyman series lies in the ultraviolet region, while the Paschen series is in the infrared region. Understanding these regions helps identify where each series appears in the spectrum.