Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problema 30

Capitolo 39, Problema 30

Use Balmer's formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the Hg line of the Balmer series for hydrogen.

Guida verificata passo dopo passo

Step 1: Understand Balmer's formula. Balmer's formula is used to calculate the wavelength of spectral lines in the hydrogen atom. It is given by:

λ_{n} = \frac{R (n^{2})}{n^{2} - m^{2}}

, where R is the Rydberg constant (approximately 1.097 × 10⁷ m⁻¹), n is the principal quantum number of the final energy level, and m is the principal quantum number of the initial energy level. For the Balmer series, m = 2.

Step 2: Identify the quantum numbers for the Hg line in the Balmer series. The Hg line corresponds to the transition where the electron moves from n = 4 to m = 2. Substitute these values into Balmer's formula to calculate the wavelength:

λ_{n} = \frac{R (4^{2})}{4^{2} - 2^{2}}

Step 3: Once the wavelength is determined, calculate the frequency using the relationship between wavelength and frequency:

f = \frac{c}{λ}

, where c is the speed of light (approximately 3 × 10⁸ m/s). Substitute the calculated wavelength into this formula to find the frequency.

Step 4: Calculate the photon energy using the formula:

E = h f

, where h is Planck's constant (approximately 6.626 × 10⁻³⁴ J·s) and f is the frequency calculated in the previous step.

Step 5: Summarize the results. You now have the wavelength, frequency, and photon energy for the Hg line of the Balmer series. Ensure all units are consistent and verify the calculations for accuracy.

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Balmer's Formula

Balmer's formula describes the wavelengths of the spectral lines in the hydrogen atom's Balmer series. It is given by the equation λ = b(n²)/(n² - 2²), where λ is the wavelength, b is a constant (approximately 364.50682 nm), and n is the principal quantum number (n = 3, 4, 5, ...). This formula allows us to calculate the wavelengths of light emitted when electrons transition from higher energy levels to the second energy level.

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Frequency and Wavelength Relationship

The frequency (ν) of a wave is inversely related to its wavelength (λ) through the equation c = νλ, where c is the speed of light (approximately 3.00 x 10^8 m/s). This relationship indicates that as the wavelength increases, the frequency decreases, and vice versa. Understanding this concept is essential for calculating the frequency of the emitted light from the wavelength obtained using Balmer's formula.

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Photon Energy

The energy (E) of a photon is directly proportional to its frequency and is given by the equation E = hν, where h is Planck's constant (approximately 6.626 x 10^-34 J·s). This relationship shows that higher frequency light corresponds to higher energy photons. Calculating the photon energy for the emitted light from the Balmer series requires knowing the frequency derived from the wavelength.

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