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Chapter 6: Problem 13

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

Short Answer

Expert verified
The wavelength of the transition from n=7 to n=5 in the Pfund series is approximately 2165.92 nm. This transition belongs to the infrared region of the electromagnetic spectrum.

Step by step solution

01

Recall the Rydberg formula

The Rydberg formula is given by: \[ \frac{1}{\lambda}=R_H \left(\frac{1}{n_{lo}^2} - \frac{1}{n_{up}^2}\right) \] where λ is the wavelength of the transition, R_H=1.097373 × 10^7 m^-1 is the Rydberg constant for hydrogen, n_lo is the lower energy level, and n_up is the higher energy level. In this problem, n_lo=5 and n_up=7.
02

Calculate the wavelength using the Rydberg formula

Insert the values of n_lo, n_up, and R_H into the Rydberg formula: \[ \frac{1}{\lambda} = 1.097373 \times 10^7 \left(\frac{1}{5^2} - \frac{1}{7^2}\right) \] Now calculate the wavelength by finding the reciprocal of the expression on the right-hand side, and convert it from meters to nanometers. \[ \lambda = \frac{1}{1.097373 \times 10^7\left(\frac{1}{5^2} - \frac{1}{7^2}\right)} \times 10^9 \] Calculate λ by plugging in the numbers: \[ \lambda \approx 2165.92 \ \text{nm} \]
03

Determine the region of the electromagnetic spectrum

Consult the electromagnetic spectrum and the wavelength of the transition: 1. Ultraviolet: 10 nm to 400 nm 2. Visible: 400 nm to 700 nm 3. Infrared: 700 nm to 1 mm Since the calculated wavelength is 2165.92 nm, it falls into the infrared region of the electromagnetic spectrum.
04

Answer

(a) The wavelength of the transition from n=7 to n=5 in the Pfund series is approximately 2165.92 nm. (b) These lines are formed in the infrared region of the spectrum.

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