According to the equation for the Balmer line spectrum of...

According to the equation for the Balmer line spectrum of hydrogen, a value of $n=3$ gives a red spectral line at $656.3 \mathrm{nm},$ a value of $n=4$ gives a green line at $486.1 \mathrm{nm},$ and a value of $n=5$ gives a blue line at 434.0 nm. Calculate the energy in kilojoules per mole of the radiation corresponding to each of these spectral lines.

According to the equation for the Balmer line spectrum of hydrogen, a value of $n=3$ gives a red spectral line at $656.3 \mathrm{nm},$ a value of $n=4$ gives a green line at $486.1 \mathrm{nm},$
and a value of $n=5$ gives a blue line at 434.0 nm. Calculate the energy in kilojoules per mole of the radiation corresponding to each of these spectral lines.

Key Concepts

Photon Energy and Unit Conversion

Calculating energy per mole of photons involves first determining the energy of an individual photon using Planck’s equation and then converting that microscopic energy to macroscopic energy using Avogadro’s number. This process, along with appropriate unit conversions (from joules to kilojoules), bridges the gap between quantum-scale events and bulk measurements commonly used in chemical and physical calculations.

Avogadro’s Number and Molar Quantities

Avogadro's number represents the number of entities in one mole, allowing for the conversion of the energy of a single photon to the total energy contained in a mole of photons. This concept is essential for translating atomic-scale energy events into quantities that are relevant in practical chemistry and physics measurements.

Planck’s Equation

Planck's equation is fundamental in quantum mechanics and relates the energy of a photon to its frequency or wavelength. The relation, E = h*c/?, where h is Planck's constant, c is the speed of light, and ? is the wavelength, forms the basis for calculating the energy of electromagnetic radiation. This concept is crucial for linking the quantized nature of light with measurable energy values.

Hydrogen Emission Spectrum

The hydrogen emission spectrum, particularly the Balmer series, illustrates the discrete wavelengths of light emitted when electrons in a hydrogen atom transition from higher energy levels to a lower energy level (n=2). Each transition produces a photon with a specific energy and wavelength, corresponding to particular colors in the visible spectrum. This concept explains the origin of the observed spectral lines.

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