Question

According to the Bohr model, the radius of a circular orbit is given by the equation $$ r(\text { in } \mathrm{nm})=0.0529 \mathrm{n}^{2} $$ Draw successive orbits for the hydrogen electron at \(\mathbf{n}=1,2\), \(3,\) and \(4 .\) Indicate by arrows transitions between orbits that lead to lines in the (a) Lyman series \(\left(\mathbf{n}_{l o}=1\right)\) (b) Balmer series \(\left(\mathrm{n}_{\mathrm{l}}=2\right)\)

Short answer

The orbital radii for the first four energy levels of a hydrogen atom are as follows: - n=1: r = 0.0529 nm - n=2: r = 0.2116 nm - n=3: r = 0.4761 nm - n=4: r = 0.8464 nm In a diagram, the energy levels can be represented as orbits of a hydrogen electron at these specific radii. To represent the Lyman series (n_lower = 1), draw arrows for transitions between higher levels (n_higher = 2, 3, 4) to the lower level of n = 1. For the Balmer series (n_lower = 2), draw arrows for transitions from higher levels (n_higher = 3, 4, and more) to the lower level of n=2. Label the arrows with their corresponding emission line names (Hα, Hβ, Hγ, etc.) for both series.

Step-by-step solution

Calculate the orbital radii

Using the equation provided above, calculate the orbital radii for the following energy levels: - n=1: r = 0.0529(1)^2 = 0.0529 nm - n=2: r = 0.0529(2)^2 = 0.2116 nm - n=3: r = 0.0529(3)^2 = 0.4761 nm - n=4: r =0.0529(4)^2 = 0.8464 nm

Draw the orbits

Using a compass or a drawing software, draw circles representing the orbits of the hydrogen electron at energy levels n=1, 2, 3, and 4. Each orbit should have its radius calculated in Step 1.

Indicate the Lyman series transitions

For the Lyman series, the lower energy level, n_lower = 1. Draw arrows that represent the allowed transitions between energy levels with n_higher = 2, 3, and 4, transitioning to n_lower = 1. Label each arrow with its corresponding emission line (Lyman series, Hα, Hβ, Hγ, ...).

Indicate the Balmer series transitions

For the Balmer series, the lower energy level, n_lower = 2. Draw arrows that represent the allowed transitions between energy levels with n_higher = 3, 4, and higher (you can stop at n=6) to n_lower = 2. Label each arrow with its corresponding emission line (Balmer series, Hα, Hβ, Hγ, ...). After completing these steps, you will have a diagram showing the orbits of the hydrogen electron at different energy levels and the transitions that result in the emission lines for the Lyman and Balmer series.

Key concepts

Orbital Radii Calculation in the Bohr Model

In the Bohr model of the hydrogen atom, electrons orbit the nucleus much like planets orbit the sun, but with quantized energy levels. To visualize this, we calculate the orbital radii where these electrons can exist without radiating energy. The equation used to determine an electron's orbital radius is a fundamental aspect of the Bohr model, represented by the formula \( r(nm) = 0.0529 n^2 \), where \( n \) is the principal quantum number corresponding to the orbit's energy level.

The orbital radius increases with the square of the energy level, meaning an electron in a higher energy orbit, such as \( n=4 \), is further from the nucleus compared to an electron in a lower energy orbit, like \( n=1 \). Knowing this, we calculate that the first four orbits have radii of 0.0529 nm, 0.2116 nm, 0.4761 nm, and 0.8464 nm, respectively. These calculations give us a concise picture of the atom's structure and how the distance from the nucleus increases significantly with each energy level.

Understanding the calculation of orbital radii allows you to appreciate the scale of atomic structures and sets the stage for interpreting spectral emissions, as electrons dropping to lower energy orbits emit specific wavelengths of light, forming the essence of the Lyman and Balmer series transitions.

Lyman Series Transitions

The Lyman series transitions are a set of spectral lines in the ultraviolet region that originates from when an excited electron falls to the atom's ground state, which is the first energy level \( (n=1) \). These transitions are fascinating for scientists because they provide valuable information about the energy levels in an atom and the nature of the electron's movement.

For instance, when an electron transitions from the second energy level \( (n=2) \), down to the first, the Lyman-alpha (Ly \( \alpha \) ) line is produced. Similarly, transitions from third \( (n=3) \) or higher energy levels down to the first will produce other lines in the Lyman series such as Lyman-beta (Ly \( \beta \) ) and Lyman-gamma (Ly \( \gamma \) ). Each of these transitions represents a photon being released with a distinctive energy or, equivalently, a unique wavelength characteristic of the energy gap being bridged.

Remember, the greater the energy difference between the levels, the more energy the photon will carry, and hence, the shorter its wavelength will be. Since the Lyman series lies in the ultraviolet range, these transitions cannot be seen with the naked eye but are crucial for understanding quantum mechanics and have been pivotal in the development of atomic theory.

Balmer Series Transitions

The Balmer series transitions are well-studied because they are visible to the human eye as various colors of light. These transitions occur when an electron moves from a higher energy level (such as \( n=3 \) or above) to the second energy level (\( n=2 \)). Each transition within this series creates a specific color, due to the particular amount of energy released as light.

An electron falling from the third energy level to the second generates the famous H-alpha (\( H\alpha \)) line, which is a red color and is often used in astronomy to identify hydrogen gas in galaxies and nebulae. Other transitions, such as from \( n=4 \) to \( n=2 \) (H-beta (\( H\beta \))), produce light in the blue-green part of the spectrum, and continue as such with various colors appearing as the initial energy level increases before the electron makes the transition.

The comprehensibility of the Balmer series in visible light makes it an excellent tool for educational demonstrations and for scientists to study the interstellar medium. By analyzing the emitted light from these transitions, they discern the physical conditions of faraway celestial objects. The simplicity of the Bohr model adds to its pedagogical value, laying a solid foundation for students to transition into more complex atomic theories.