Question

In which region of electromagnetic spectrum does the Lyman series of hydrogen atom like (A) \(x\) -ray (B) Infrared (C) visible (D) ultraviolet

Short answer

The Lyman series of hydrogen atom-like falls in the (D) Ultraviolet region of the electromagnetic spectrum. The smallest and largest transitions in the Lyman series have wavelengths of approximately 36.46 nm and 91.13 nm, respectively, which both lie within the ultraviolet range (10 nm to 400 nm).

Step-by-step solution

Review the Rydberg Formula for Hydrogen-like Atoms

The Rydberg formula for hydrogen-like atoms is used to calculate the wavelengths of the spectral lines in the electromagnetic spectrum. The formula is as follows: \[\frac{1}{\lambda} = R Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\] Where \(λ\) is the wavelength, \(R\) is the Rydberg constant (approx. 1.097 x \(10^7 m^{-1}\)), \(Z\) is the atomic number of the hydrogen-like atom (for hydrogen, Z = 1), \(n_1\) is the lower energy level, and \(n_2\) is the higher energy level.

Calculate the Wavelength for the Lyman Series Transitions

Since the Lyman series represents transitions where the lower energy level (n1) is 1 (ground state), we will simplify the Rydberg formula for the Lyman series: \[\frac{1}{\lambda} = R \left(\frac{1}{1^2} - \frac{1}{n_2^2}\right)\] Now, let's find the wavelength for the smallest and largest transition within the Lyman series. The smallest transition occurs when n_2 = 2 (from upper energy level 2 to the ground state). The largest transition occurs when n_2 approaches infinity. For the smallest transition, use the formula: \[\frac{1}{\lambda_{smallest}} = R \left(\frac{1}{1^2} - \frac{1}{2^2}\right) = R \left(1 - \frac{1}{4}\right) = \frac{3}{4} R\] Now, the wavelength for the smallest transition is calculated as: \[\lambda_{smallest} = \frac{1}{\frac{3}{4} R}\] For the largest transition, n_2 approaches infinity, and the formula becomes: \[\frac{1}{\lambda_{largest}} = R \left(\frac{1}{1^2} - 0\right) = R\] Now, the wavelength for the largest transition is calculated as: \[\lambda_{largest} = \frac{1}{R}\]

Determine the Region of the Electromagnetic Spectrum

Now that we have found the wavelength range for the Lyman series, we can determine in which region of the electromagnetic spectrum it belongs. The boundaries for the different regions of the spectrum are: - X-ray: less than 10 nm - Ultraviolet (UV): 10 nm to 400 nm - Visible: 400 nm to 700 nm - Infrared (IR): greater than 700 nm Plug the value of Rydberg constant in the formulas and find the values of smallest and largest wavelength transition: \(λ_{smallest} = 4 \times (1.097 \times 10^7)^{-1} \approx 3.646 \times 10^{-8}\:m = 36.46\:nm \) \(λ_{largest} = (1.097 \times 10^7)^{-1} \approx 9.113 \times 10^{-8}\:m = 91.13\:nm \) Since the smallest and largest transitions in the Lyman series lie within the ultraviolet region of the electromagnetic spectrum, the answer is: (D) Ultraviolet