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The Balmer series is a sequence of spectral lines emitted by hydrogen atoms as electrons transition from higher energy levels down to the second energy level, known as n = 2. To calculate the wavelength for the \( \mathrm{H}_{\gamma} \) line, which corresponds to a transition where the electron falls from n = 5 to n = 2, we use a specific formula known as Balmer's formula. This formula is critical in determining the wavelengths of the lines in the Balmer series.
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \]
In this formula, \( \lambda \) is the wavelength, \( R \) is the Rydberg constant \( (1.097373 \times 10^7 \, \mathrm{m}^{-1}) \), and \( n \) is the principal quantum number of the final energy level, which is 5 in this case.

• The equation simplifies for our specific case to: \[ \frac{1}{\lambda} = R \left( \frac{1}{4} - \frac{1}{25} \right) \]
• Solving for \( \lambda \), we first calculate the quantity in the brackets: \( \frac{1}{4} - \frac{1}{25} = \frac{21}{100} \).
• We then find lambda by inverting this value and multiplying by the Rydberg constant \( R \).

The calculated wavelength is the one that characteristically describes the \( \mathrm{H}_{\gamma} \) line in the visible spectrum of hydrogen.

Once the wavelength \( (\lambda) \) is determined using Balmer's formula, we can use it to find the frequency \( (u) \) of the light emitted. Frequency is crucial for understanding how often a wave oscillates per second. This can be calculated using the formula that relates speed, wavelength, and frequency: \[ c = \lambda u \]
where \( c \) represents the speed of light \( (2.998 \times 10^8 \, \mathrm{ms}^{-1}) \). Rearranging this equation to solve for \( u \) gives us: \[ u = \frac{c}{\lambda} \]

• After calculating \( \lambda \) in the previous step, plug it into this formula.
• The result is the frequency of the \( \mathrm{H}_{\gamma} \) line, usually expressed in hertz (Hz).

Frequency informs us how many times the wave oscillates in one second and is directly related to the energy and color of the light.

The energy of a photon is directly related to its frequency, and this relationship is described by Planck's equation. The energy of a photon is crucial in quantum mechanics as it explains how light transfers energy.
The formula for calculating photon energy is: \[ E = h u \]
where \( E \) is the energy of the photon, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \mathrm{Js}) \), and \( u \) is the frequency we calculated in the previous section.

• To find the energy, multiply \( h \) by \( u \).
• This result gives the energy in joules, a measure of how much energy each photon of the \( \mathrm{H}_{\gamma} \) line carries.

Understanding photon energy is essential for explaining the photoelectric effect and other phenomena. This energy is what allows photons to interact with matter, impacting how we perceive light.