Problem 53 - College Physics

Question

(a) What is the least amount of energy, in electronvolts, that must be given to a hydrogen atom that is initially in its ground level so that it can emit the \(\mathrm{H}_{\alpha}\) line in the Balmer series? (b) How many different possibilities of spectral-line emissions are there for this atom when the electron starts in the \(n=3\) level and eventually ends up in the ground level? Calculate the wavelength of the emitted photon in each case.

Step-by-Step Solution

Answer

(a) 12.09 eV needed. (b) Three emissions: 656 nm, 102.8 nm, 121.6 nm.

Step 1: Understanding the Problem

For part (a), to emit the \( \mathrm{H}_{\alpha} \) line in the Balmer series, the electron must transition from the \( n=3 \) to \( n=2 \) state. For part (b), the problem is asking for all possible spectral emissions starting from \( n=3 \) and ending at \( n=1 \).

Step 1: Calculate Energy for Part (a)

The \( \mathrm{H}_{\alpha} \) line corresponds to the transition from \( n=3 \) to \( n=2 \). First, calculate the energy difference between levels using \( E_n = - \frac{13.6}{n^2} \) eV. \( E_3 = -\frac{13.6}{9} \) eV and \( E_2 = -\frac{13.6}{4} \) eV. The energy required to initially excite the atom from \( n=1 \) to \( n=3 \) must be calculated as \( E_3 - E_1 \).

Step 2: Energy Difference Calculation

Calculate \( E_1 = -13.6 \) eV. Thus, the energy needed to excite from ground state to \( n=3 \) is \( E_3 - E_1 = -\frac{13.6}{9} - (-13.6) \) eV, which simplifies to \( \frac{121.6}{9} \approx 13.6 \times 0.8889 \approx 12.09 \) eV.

Step 3: Calculate Wavelength of \( \mathrm{H}_{\alpha} \) Transition

Use the energy difference between \( n=3 \) and \( n=2 \), \( E_{32} = \frac{13.6}{4} - \frac{13.6}{9} \approx 1.89 \) eV. The photon wavelength \( \lambda \) is given by \( \lambda = \frac{hc}{E} \), where \( h \) is Planck's constant and \( c \) is the speed of light.

Step 4: Calculate Possible Transitions for Part (b)

From \( n=3 \), the electron can transition to \( n=2 \) then to \( n=1 \), or directly from \( n=3 \) to \( n=1 \). There are two possible direct emissions: \( n=3 \rightarrow n=2 \) and \( n=3 \rightarrow n=1 \), plus one indirect resulting emission: \( n=2 \rightarrow n=1 \).

Step 5: Calculate Wavelengths for Part (b)

1) \( n=3 \rightarrow n=2 \): \( E_{32} \approx 1.89 \) eV, \( \lambda \approx 656 \) nm. 2) \( n=3 \rightarrow n=1 \): \( E_{31} \approx 12.09 \) eV, \( \lambda \approx 102.8 \) nm. 3) \( n=2 \rightarrow n=1 \): \( E_{21} = 10.2 \) eV, \( \lambda \approx 121.6 \) nm.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest element, consisting of just one proton and one electron. This simplicity makes it an excellent candidate for studying quantum mechanics and atomic transitions. When dealing with the hydrogen atom, it's important to remember that the electron is situated in distinct energy levels or orbits, commonly labeled by the quantum number \( n \). These energy levels range from the ground state \( n=1 \) to higher excited states. Each step up corresponds to a greater potential energy level for the electron.
The behavior of the electron in a hydrogen atom forms the foundation of spectroscopy and aids in understanding more complex atomic structures. As the electron transitions between these energy levels within the atom, it emits or absorbs specific amounts of energy, giving rise to spectral lines.

Electron Transitions

Electron transitions in a hydrogen atom refer to the movement of the electron between different energy levels. These transitions occur when energy is absorbed or released by the electron. For instance, if an electron receives energy, it can "jump" from a lower to a higher energy level, known as excitation. Conversely, when it falls from a higher to a lower energy level, it emits energy, often in the form of a photon. This photon is precisely what we observe as spectral lines.
Understanding these transitions is essential for explaining phenomena such as the Balmer series, where electrons fall to the \( n=2 \) level from higher levels. Notably, the energy difference between the initial and final levels determines the frequency and wavelength of the emitted photon during these transitions.

Spectral Lines

Spectral lines are the result of electron transitions between energy levels in an atom, such as the hydrogen atom. Each spectral line corresponds to a specific wavelength of light and is a unique fingerprint of the element. When electrons in a hydrogen atom transition from one energy level to another, they emit or absorb photons, resulting in spectral lines that can be observed in a spectrum.
The Balmer series is a set of spectral lines associated with electron transitions from higher energy levels to the \( n=2 \) energy level in hydrogen. These transitions produce visible lines, such as the \( \mathrm{H}_{\alpha} \) line, and help in identifying hydrogen in various cosmic and laboratory environments.

Electronvolts

Electronvolts (eV) are a unit of energy that's particularly useful in the context of atomic physics and electron transitions. One electronvolt is the amount of kinetic energy gained by a single electron when it is accelerated through an electric potential difference of one volt. In terms of practical measurements, \( 1 \, \mathrm{eV} = 1.602 \times 10^{-19} \) joules.
In atomic transitions, such as those in the hydrogen atom, electronvolts offer a convenient way to quantify the energy differences between various energy levels. For example, when calculating the energy required for an electron to transition between the \( n=1 \) ground state and higher levels, the calculations are often expressed in electronvolts for simplicity and precision.

Photon Wavelength

Photon wavelength is a crucial concept when discussing spectral lines and electron transitions. The wavelength of a photon is inversely related to its energy, as given by the equation \( \lambda = \frac{hc}{E} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( E \) is the energy of the photon. Therefore, the greater the energy of the photon, the shorter its wavelength.
In the context of the hydrogen atom, when electrons transition between different energy levels, they emit photons with specific wavelengths that correspond to the spectral lines observed. For example, during the \( \mathrm{H}_{\alpha} \) transition in the Balmer series, a photon with a wavelength of approximately 656 nm is emitted, which falls within the visible spectrum.