Problem 17 - Chemistry and Chemical Reactivity

Question

Consider only transitions involving the \(n=1\) through \(n=5\) energy levels for the H atom (where the energy level spacings below are not to scale). $$\begin{aligned} &\begin{aligned} & n=5 \\ & n=4 \\ & n=3 \\ & n=2 \end{aligned}\\\ &7\\\ &n=1 \end{aligned}$$ (a) How many emission lines are possible, considering only the five quantum levels? (b) Photons of the highest frequency are emitted in a transition from the level with \(n=\quad\) to a level with \(n=\) (c) The emission line having the longest wavelength corresponds to a transition from the level with \(n=\ldots\) to the level with \(n=\)

Step-by-Step Solution

Answer

(a) 10 lines. (b) \( n=5 \) to \( n=1 \). (c) \( n=2 \) to \( n=1 \).

Step 1: Calculate Possible Emission Lines

To find the number of possible emission lines for transitions involving energy levels from \( n=1 \) to \( n=5 \), we use the formula for the sum of an arithmetic series. Every energy level \( n \) can transition to any lower level \( m \) (where \( m < n \)).This yields the series of transitions:- From \( n=5 \): transitions to \( n=4, 3, 2, 1 \) \( \Rightarrow 4 \) lines- From \( n=4 \): transitions to \( n=3, 2, 1 \) \( \Rightarrow 3 \) lines- From \( n=3 \): transitions to \( n=2, 1 \) \( \Rightarrow 2 \) lines- From \( n=2 \): transition to \( n=1 \) \( \Rightarrow 1 \) lineAdding them up gives a total of \( 4+3+2+1 = 10 \) possible emission lines.

Step 2: Determine Highest Frequency Transition

The photon of the highest frequency results from the transition with the greatest energy change. Energy in the hydrogen atom levels is inversely related to the square of the principal quantum number \( n \), given by the formula:\[ E_n = - \frac{13.6 \text{ eV}}{n^2} \]A transition from the highest initial energy (\( n=5 \)) to the lowest final energy (\( n=1 \)) gives the largest energy change:\[ \text{Transition: } n=5 \rightarrow n=1 \]

Step 3: Determine Longest Wavelength Transition

The longest wavelength corresponds to the smallest energy transition (as energy and wavelength are inversely related):\[ E = \frac{hc}{\lambda} \Rightarrow \lambda \propto \frac{1}{\Delta E} \]The smallest energy transition occurs between consecutive energy levels. For the levels available, this is the transition from \( n=2 \rightarrow n=1 \): this is the situation where the energy difference between transmitted photons is minimal \( \Rightarrow \text{Transition: } \ n=2 \rightarrow n=1 \).

Key Concepts

Quantum Levels

In quantum mechanics, electrons orbit around atoms within specific regions referred to as quantum levels or energy levels. For the hydrogen atom, these levels are indicated by integer numbers, known as "n". These numbers range from 1 to infinity, but in most cases, we only consider a few low lying levels.
These quantum levels indicate the potential energy the electron possesses, within the atom, due to its position. The lower the level number, the closer the electron is to the nucleus and the lower its energy. Hence, when an electron transitions between these levels, energy is either absorbed or emitted in the form of light.

• The ground state is when the electron is in the lowest energy level, which is often called the 'n=1'.
• Higher levels (like n=2 to n=5) are considered excited states, and electrons transitioning to these states involve absorbing energy.

Understanding these levels is crucial for determining possible transitions and the respective emitted or absorbed light.

Energy Transition

When an electron moves from one quantum level to another, it undergoes an energy transition. These transitions are quantified by the difference in energy levels of the electron's initial and final states.
For the hydrogen atom, this is described mathematically by the formula: \[ E_n = - \frac{13.6 \text{ eV}}{n^2} \]Here, the energy is expressed in electron volts (eV), and you can observe that energy becomes less negative, or increases, as "n" increases. Thus, the amount of energy emitted or absorbed is equal to the difference between these energy levels.

• A transition to a lower energy level emits energy, releasing light.
• A move to a higher energy level requires energy absorption, using light.

The larger the jump between levels, the more energy is involved in the transition, affecting the characteristics of the emitted or absorbed light, such as its color.

Wavelength and Frequency

Energy transitions in hydrogen lead to the emission or absorption of photons. Photons have properties such as wavelength and frequency, two fundamental characteristics of light. The relationship between the energy of a photon and these properties is given by:\[ E = h u = \frac{hc}{\lambda} \]Here, \( E \) is energy, \( h \) is Planck's constant, \( u \) is frequency, \( c \) is the speed of light, and \( \lambda \) is wavelength.
The energy is directly proportional to frequency, implying that a photon with higher energy will have a higher frequency and a shorter wavelength. Conversely, photons with less energy have lower frequencies and longer wavelengths.

• Higher frequency light appears towards the violet end of the electromagnetic spectrum.
• Lower frequency light appears towards the red end of the spectrum.

Thus, the frequency and wavelength give clues about the energy changes occurring within the atom.

Photons and Energy Levels

Photons are the particles of light that carry energy during transitions between energy levels in an atom. When an electron falls from a higher energy level to a lower one, it emits a photon, and its energy is equal to the difference in energy between the two levels.
To calculate the energy of the photon, you can use the formula:\[ \Delta E = E_{final} - E_{initial} \]Where \( \Delta E \) represents the change in energy. The emitted photon's energy corresponds to this change.

• Photons with high energy come from large energy transitions and are seen as high-frequency, short-wavelength radiation.
• Conversely, photons with lower energy arise from smaller energy transitions and appear as low-frequency, longer-wavelength radiation.

Thus, examining the photon's energy and associated characteristics can provide insights into the atomic processes taking place during these transitions.