Question

For the Pfund series, \(\mathbf{n}_{\mathrm{lo}}=5\). (a) Calculate the wavelength in nanometers of a transition from \(\mathrm{n}=7\) to \(\mathrm{n}=5\) (b) In what region of the spectrum are these lines formed?

Short answer

Question: Determine the wavelength of the emitted photon when an electron transitions from n=7 to n=5 in a hydrogen atom, and identify the region of the electromagnetic spectrum where this wavelength corresponds. Answer: (a) The wavelength of the transition from n=7 to n=5 is approximately 2179.7 nm. (b) These lines are formed in the infrared region of the electromagnetic spectrum.

Step-by-step solution

Identify the variables in the Rydberg Formula

The Rydberg formula for calculating the wavelength of emitted or absorbed photons for a hydrogen-like atom is: \( \frac{1}{\lambda} = R_H * Z^2 * (\frac{1}{n_1^2} - \frac{1}{n_2^2}) \) Where: - \(\lambda\) is the wavelength of the emitted/absorbed photon - \(R_H\) is the Rydberg constant for hydrogen, approximately 1.097 * 10^7 m^-1 - \(Z\) is the atomic number (for hydrogen, Z=1) - \(n_1\) is the lower principal quantum number (n_lo) - \(n_2\) is the upper principal quantum number (n_hi) In this exercise, we are asked to find the wavelength when \(\mathbf{n}_{\mathrm{lo}}=5\), and \(\mathrm{n}=7\).

Calculate the Wavelength

Using the Rydberg formula, we'll calculate the wavelength of the emitted photon when an electron transitions from n=7 to n=5: \( \frac{1}{\lambda} = R_H * Z^2 * (\frac{1}{n_1^2} - \frac{1}{n_2^2}) \) \( \frac{1}{\lambda} = (1.097 * 10^7 m^{-1}) * (1^2) * (\frac{1}{5^2} - \frac{1}{7^2}) \) Now, solve for the wavelength: \( \lambda = \frac{1}{1.097 * 10^7 * (\frac{1}{25} - \frac{1}{49})} \) \( \lambda \approx 2.1797 * 10^{-6} m \) To convert the wavelength to nanometers, multiply by \(10^9\): \( \lambda \approx 2179.7 \ nm \)

Identify the Region of the Spectrum

The calculated wavelength is approximately 2179.7 nm, which falls within the infrared region of the electromagnetic spectrum.

Answer

(a) The wavelength of the transition from n=7 to n=5 is approximately 2179.7 nm. (b) These lines are formed in the infrared region of the electromagnetic spectrum.