Problem 14 - General Chemistry: Principles and Modern Applications

Question

What is \(\Delta E\) for the transition of an electron from \(n=5\) to \(n=2\) in a Bohr hydrogen atom? What is the frequency of the spectral line produced?

Step-by-Step Solution

Answer

To find \(\Delta E\), calculate the energies at each state, and subtract \(E_5\) from \(E_2\). To find the frequency, use the derived \(\Delta E\) value and apply the equation \(E=h\nu\). Remember to convert \(\Delta E\) from eV to Joules before calculating the frequency.

Step 1: Calculation of energies in different orbitals

Calculate the energies in both states using the equation \(E=-13.6/n^2\). Substitute \(n=5\) to find \(E_5\), and \(n=2\) to find \(E_2\). The energies are in electron volts (eV).

Step 2: Calculation of the Energy difference

Determine the difference in energy levels. The energy difference \(\Delta E = E_2 - E_5\).

Step 3: Convert energy difference to Joules

The resulting energy difference will be in electron volts. Convert this value to joules using the conversion factor \(1 eV = 1.6 x 10^{-19} J \) to use in the next step.

Step 4: Calculation of spectral line frequency

Having gotten \(\Delta E\) in joules, use it to calculate the frequency of the spectral line produced using the relation \(E=h\nu\), where \(h\) is Planck's constant \((6.62607004 × 10^-34 m^2 kg / s)\). Solve for \(\nu\) by dividing \(\Delta E\) by \(h\)

Key Concepts

Energy Level Transition

In the Bohr model of the hydrogen atom, electrons transition between energy levels which are quantized. Each level is defined by a principal quantum number, denoted by \( n \). The energy of an electron in the nth level is given by the formula: \[ E_n = \frac{-13.6}{n^2} \text{ eV} \]Here, energy (E) is in electron volts (eV) and 13.6 is the ionization energy of hydrogen from the ground state (n=1).

• A transition involves moving from one energy level to another.
• Energy levels with higher values of \( n \) are higher in energy but closer together.
• Every energy level is negative, indicating bound states compared to a free electron.

When an electron drops from a higher to a lower energy level, it releases energy. The energy difference \( \Delta E \) between two levels, such as from \( n=5 \) to \( n=2 \), is calculated by substituting the respective \( n \) values into the formula above and finding the difference:\[ \Delta E = E_{2} - E_{5} \]This energy difference represents the amount of energy emitted as the electron transitions, visible as a spectral line.

Spectral Line Frequency

Spectral lines refer to the specific wavelengths of light emitted or absorbed by electrons transitioning between energy levels. The frequency \( u \) of a spectral line represents how many wave cycles occur per second and is linked to the energy change \( \Delta E \) by the equation:\[ \Delta E = h u \]where:

• \( \Delta E \) is the energy difference (already converted to joules).
• \( h \) is Planck's constant \( 6.62607004 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \).

To find the frequency, solve for \( u \):\[ u = \frac{\Delta E}{h} \]This frequency is often in the visible or ultraviolet range for hydrogen.
The conversion of \( \Delta E \) to joules is crucial because it aligns the unit with \( h \), ensuring the resulting frequency is in hertz \( \text{Hz} \). Measuring this frequency allows scientists to identify elements and study their properties via spectral lines.

Electron Volts to Joules Conversion

Understanding the conversion between electron volts (eV) and joules (J) is crucial for practical calculations in physics and chemistry, particularly when dealing with atomic and subatomic energies.

• 1 electron volt \( (eV) \) is the amount of kinetic energy gained or lost by an electron as it moves through an electric potential difference of one volt.
• It is a convenient unit for describing atomic-scale energies.
• However, calculations involving energy changes, like spectral lines, require SI units, which means energies must be converted into joules.

The conversion factor is:\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]To convert an energy value from eV to J, simply multiply by this factor.
Example: If \( \Delta E \) in a given transition is 3 eV, converting to joules involves:\[ \Delta E = 3 \times 1.6 \times 10^{-19} = 4.8 \times 10^{-19} \text{ J} \]This conversion permits calculations to proceed seamlessly with other fundamental constants, enabling accurate computation of spectral frequencies and facilitating deeper study in quantum mechanics.