Question

When electron jumps from the fourth orbit to the second orbit in \(\mathrm{He}^{+}\) ion, the radiation emitted out will fall in (a) ultraviolet region (b) visible region (c) infrared region (d) radio wave region

Short answer

The photon emitted during the transition from the fourth to the second orbit of the \(\mathrm{He}^{+}\) ion falls in the ultraviolet region.

Step-by-step solution

Recall the energy levels in a hydrogen-like atom

In a hydrogen-like atom such as \(\mathrm{He^{+}}\), the energy levels can be calculated using the formula \(E_n = -\frac{Z^2 R}{n^2}\), where \(E_n\) is the energy of the nth orbit, \(Z\) is the atomic number, \(R\) is the Rydberg constant for hydrogen, and \(n\) is the principal quantum number. For \(\mathrm{He^{+}}\), \(Z = 2\).

Determine the energy difference between the orbits

Calculate the energy of the fourth orbit \(E_4\) and the second orbit \(E_2\), then find the difference \(\Delta E = E_2 - E_4\). Use the formula \(E_n = -\frac{Z^2 R}{n^2}\) for both orbits, substituting \(n=2\) and \(n=4\), and \(Z=2\) for \(\mathrm{He}^{+}\). This energy difference corresponds to the energy of the photon emitted when an electron transitions from the fourth to the second orbit.

Identify the region of the electromagnetic spectrum

The energy difference calculated in Step 2 will give the energy of the emitted photon. By using the energy relation \(E = hu\), where \(h\) is Planck's constant and \(u\) is the frequency of the radiation, determine the frequency. With the frequency known, refer to the electromagnetic spectrum to identify the region (ultraviolet, visible, infrared, radio wave) in which this frequency falls.

Key concepts

Rydberg Constant for Hydrogen

The Rydberg constant for hydrogen is a fundamental physical constant crucial for understanding the energy levels of electrons in a hydrogen atom. It is represented by the symbol \( R \) and has a value of approximately \( 1.097 \times 10^7 \) per meter (\( m^{-1} \)).

When an electron transitions between orbits in a hydrogen-like atom, the change in energy is related to the Rydberg constant, which can be used to calculate the wavelengths or frequencies of the photons absorbed or emitted during these transitions. The formula incorporating the Rydberg constant is:
\[ E_n = -\frac{Z^2 \times R}{n^2} \]
where \( E_n \) is the energy of the nth orbit, \( Z \) is the atomic number specific to the atom, and \( n \) is known as the principal quantum number. For a \textbf{hydrogen-like atom}, which includes ions with one electron such as \( \text{He}^+ \), \( Z \) represents the number of protons in the nucleus, and its value becomes crucial in calculating the energy levels.

Understanding the relationship between energy levels and the Rydberg constant is essential for solving problems involving electron transitions and resultant photon emissions. This is particularly useful in spectroscopy for identifying elements based on their spectral lines.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is an integral value starting from 1 that specifies the energy level or shell of an electron in an atom. In the context of hydrogen-like atoms, which include any single-electron ions, the value of \( n \) correlates directly with the size of the orbit and the energy an electron has.

The energy levels of electrons are quantized, meaning that an electron can only exist in specific allowed orbits or levels, represented by \( n \). The energy associated with each level is given by the formula \( E_n = -\frac{Z^2 \times R}{n^2} \). As \( n \) increases, the energy of the orbit becomes less negative, indicating electrons at higher levels are less tightly bound to the nucleus, hence more easily removed.

When an electron moves between energy levels, it either absorbs or emits energy. The magnitude of this energy change is often measured using the photon's frequency or wavelength produced or absorbed. The larger the jump in principal quantum number (such as from \( n=4 \) to \( n=2 \)), the greater the energy exchange. This concept is pivotal when predicting spectral lines and understanding the emission or absorption spectra.

Electromagnetic Spectrum

The electromagnetic (EM) spectrum encompasses all types of electromagnetic radiation, arranged according to frequency or wavelength. It includes, from shortest wavelength to longest: gamma rays, X-rays, ultraviolet (UV) rays, visible light, infrared (IR) radiation, microwaves, and radio waves.

Each type of electromagnetic radiation within the spectrum carries different energies. Photons of light in the UV range have more energy than those in the visible range, which in turn are more energetic than IR photons. When electrons in atoms transition between different energy levels, they emit or absorb photons that correspond to specific regions of the EM spectrum, depending on the energy difference between the levels.

For example, transitions involving higher energy differences, such as those closer to the nucleus of an atom, will result in the emission of UV or even X-ray photons, while transitions between higher energy levels with smaller differences will produce visible or infrared light. By recognizing the region of the EM spectrum that a photon belongs to, scientists can deduce key information about the energy transitions within atoms, which is essential in fields such as astrophysics, chemistry, and materials science.