Problem 195 - Introductory Chemistry Atoms First

Question

Which of the following electron transitions in hydrogen would absorb the largest amount of energy? A. \(n=2\) to \(n=3\) B. \(\quad n=2\) to \(n=4\) C. \(n=1\) to \(n=4\) D. \(n=3\) to \(n=1\) E. \(n=7\) to \(n=1\)

Step-by-Step Solution

Answer

The electron transition in hydrogen that absorbs the largest amount of energy is C: \(n = 1\) to \(n = 4\).

Step 1: Recall the energy transition formula in a hydrogen atom

The formula for the energy (E) associated with a particular energy level (principal quantum number, n) in a hydrogen atom is given by: \( E = -\dfrac{13.6\, \mathrm{eV}}{n^2} \) When an electron in a hydrogen atom undergoes a transition between two energy levels (levels \(n_i\) and \(n_f\)), the energy absorbed or emitted is given by: \( \Delta E = E_{n_f} - E_{n_i} \) This formula can be applied to each of the transitions given in the exercise.

Step 2: Calculate the energy for each transition

We will now calculate the energy absorbed for each of the given transitions using the formula mentioned earlier: A) \( \Delta E_A = E_3 - E_2 = -\dfrac{13.6\, \mathrm{eV}}{3^2} + \dfrac{13.6\, \mathrm{eV}}{2^2} \) B) \( \Delta E_B = E_4 - E_2 = -\dfrac{13.6\, \mathrm{eV}}{4^2} + \dfrac{13.6\, \mathrm{eV}}{2^2} \) C) \( \Delta E_C = E_4 - E_1 = -\dfrac{13.6\, \mathrm{eV}}{4^2} + \dfrac{13.6\, \mathrm{eV}}{1^2} \) D) \( \Delta E_D = E_1 - E_3 = -\dfrac{13.6\, \mathrm{eV}}{1^2} + \dfrac{13.6\, \mathrm{eV}}{3^2} \) E) \( \Delta E_E = E_1 - E_7 = -\dfrac{13.6\, \mathrm{eV}}{1^2} + \dfrac{13.6\, \mathrm{eV}}{7^2} \)

Step 3: Compare the energy values for each transition

Now we will compare the values of the energy absorbed for each of the transitions: \( \Delta E_A \) = \( -\dfrac{13.6\, \mathrm{eV}}{9} + \dfrac{13.6\, \mathrm{eV}}{4} > 0 \) \( \Delta E_B \) = \( -\dfrac{13.6\, \mathrm{eV}}{16} + \dfrac{13.6\, \mathrm{eV}}{4} > 0 \) \( \Delta E_C \) = \( -\dfrac{13.6\, \mathrm{eV}}{16} + \dfrac{13.6\, \mathrm{eV}}{1} > 0 \) \( \Delta E_D \) = \( -\dfrac{13.6\, \mathrm{eV}}{1} + \dfrac{13.6\, \mathrm{eV}}{9} < 0 \) \\ \( \Delta E_E \) = \( -\dfrac{13.6\, \mathrm{eV}}{1} + \dfrac{13.6\, \mathrm{eV}}{49} < 0 \) The negative values for \(\Delta E_D\) and \(\Delta E_E\) indicate that the energy is getting emitted rather than absorbed from energy transitions D and E. Therefore, we need to consider only the energy absorbed values from transitions A, B, and C. After quick observation, we can see that the largest value is \(\Delta E_C\), as it has a larger denominator in the first term.

Step 4: Identify the transition that absorbs the largest amount of energy

To identify the electron transition, we find which option corresponds to the largest energy calculation in the previous step. Hence, the electron transition in hydrogen that would absorb the largest amount of energy is C: \(n = 1\) to \(n = 4\).

Key Concepts

Energy Absorption in Electron Transitions

When an electron transitions between energy levels in an atom, the process involves either absorption or emission of energy. In hydrogen, these transitions occur as electrons move between different orbits with distinct energy levels.
The energy absorbed or emitted corresponds to the difference between the initial and final energy levels. This can be calculated using the energy transition formula: \[ \Delta E = E_{n_f} - E_{n_i} \]where:

• \( \Delta E \) is the energy change
• \( E_{n_f} \) is the final energy level
• \( E_{n_i} \) is the initial energy level

In terms of absorption, if an electron jumps to a higher energy state (higher orbit), it must absorb energy. Conversely, when it falls back to a lower state, it releases energy. These principles underlie the potential transitions analyzed in the exercise, emphasizing that only transitions leading to higher energy levels result in energy absorption.

Principal Quantum Number

The principal quantum number, usually denoted as \( n \), specifies the energy level of an electron within an atom. It is integral to understanding electron configurations because each energy level corresponds to a specific value of \( n \).
The principal quantum number can take positive integer values (\( n = 1, 2, 3, \ldots \)). As \( n \) increases, the associated energy level also increases, indicating that the electron is located further from the nucleus.
In the context of hydrogen electron transitions, knowing the principal quantum numbers involved helps to determine:

• Which kind of transition is happening
• Whether energy is absorbed or emitted as a result
• The relative difference between energy levels

The quantum number helps categorize the electron's potential orbits, impacting its stability and the likelihood of transitions occurring between levels. This concept aids in differentiating transitions based on their associated energy changes.

Energy Level Calculations

The energy level calculation forms the foundation for analyzing electron transitions in atoms, particularly in the Bohr model of hydrogen. Each energy level can be calculated using the formula:\[ E_n = -\frac{13.6 \text{ eV}}{n^2} \]where \( E_n \) is the energy of level \( n \), and \( -13.6 \text{ eV} \) is the constant specific to hydrogen's ground state energy level.
Using this formula, we can determine the energy for any given level \( n \). For electron transitions:

• Identify initial and final levels \( n_i \) and \( n_f \) respectively
• Find the energy for each using the formula
• Calculate the difference: \( \Delta E = E_{n_f} - E_{n_i} \)

This calculation tells us if energy is absorbed or emitted. In the exercise, these calculations were done for provided transitions, helping to conclude which transition results in maximum energy absorption.

Bohr Model of the Atom

The Bohr model of the atom significantly advanced our understanding by introducing quantized energy levels for electrons within atoms. It remains a simplified but vital tool for explaining the behavior of hydrogen electrons as they transition between energy levels.
In the Bohr model:

• Electrons orbit the nucleus in distinct paths called shells or energy levels
• Each level is associated with specific energy and defined by \( n \), the principal quantum number
• Energy differences between levels determine photon absorption or emission

For hydrogen, the model accurately predicts the energy levels and transitions that an electron can undertake. Its simplicity lies in treating electrons similar to planets orbiting a sun, clarifying why certain kinds of transitions are allowed or forbidden, based on changes in energy between these quantized states. In this context, it provides a backdrop for calculations that determine energy absorption.