Problem 204 - Introductory Chemistry Atoms First

Question

When a hydrogen atom is excited in a flame, a line of blue-violet light is emitted. This happens when an electron makes a transition between the \(n=5\) and the \(n=2\) orbit of the atom. If the energy of this light is \(2.9 \mathrm{eVs}\) and the energy of an electron in the \(n=2\) hydrogen orbit is \(11.2 \mathrm{eVs}\), what must be the energy of the \(n=5\) orbit?

Step-by-Step Solution

Answer

The energy of an electron in the \(n=5\) hydrogen orbit is \(14.1 \ \mathrm{eV}\).

Step 1: Analyze the given information

The energy difference between the \(n=5\) and \(n=2\) orbits is related to the energy of the emitted light. According to the problem, the emitted light has an energy of \(2.9 \ \mathrm{eV}\). The energy of the electron in the \(n=2\) orbit is given as \(11.2 \ \mathrm{eV}\).

Step 2: Relate energy difference between orbits to emitted light energy

To find the energy of the \(n=5\) orbit, we will first need to determine the energy difference between the \(n=5\) and \(n=2\) orbits. As the electron transitions from the higher energy orbit (\(n=5\)) to the lower energy orbit (\(n=2\)), the difference in energy is given off as emitted light. Therefore, the energy difference is equal to the energy of the emitted light, which is given as: Energy difference = Emitted light energy = \(2.9 \ \mathrm{eV}\)

Step 3: Determine the energy of the \(n=5\) orbit

We can now find the energy of the electron in the \(n=5\) orbit by adding the energy difference to the energy of the \(n=2\) orbit: Energy of \(n=5\) orbit = Energy of \(n=2\) orbit + Energy difference = \(11.2 \ \mathrm{eV} + 2.9 \ \mathrm{eV}\) Calculating, we get: Energy of \(n=5\) orbit = \(14.1 \ \mathrm{eV}\)

Step 4: Provide the final answer

The energy of an electron in the \(n=5\) hydrogen orbit is \(14.1 \ \mathrm{eV}\).

Key Concepts

Atomic Orbit Energy

In discussing the energy of atomic orbits, crucial to the understanding of electron transitions in chemistry, it's essential to recognize that electrons in an atom occupy certain energy levels, also known as orbits or shells. These energy levels are quantized, meaning that an electron can only exist at specific energy states and not anywhere in between.

For instance, in the hydrogen atom exercise, the energy levels of the orbits where the electron transitions are used to calculate the energy of the emitted light. The concept here hinges on the fact that when an electron 'jumps' from a higher energy orbit, like the n=5 level, to a lower one, such as the n=2 level, the energy difference is released in the form of a photon. This photon carries energy, which, in the world of quantum mechanics, is manifested as the light we can detect.

Calculating Orbit Energy

Simply put, the energy of an orbit can often be calculated by taking the known energy of a lower orbit and adding the energy of the emitted (or absorbed) photon. This calculation assumes that the energy of the photon corresponds to the energy gap between the respective orbits. As seen in our exercise, knowing the energy of the n=2 orbit and the photon allows us to compute the energy of the n=5 orbit. This is an illustration of the conservation of energy at a quantum level.

Quantum Mechanics in Chemistry

Quantum mechanics plays a pivotal role in understanding chemical processes at the atomic level. It is the branch of physics that deals with the behavior of particles on a very small scale. In the realm of chemistry, quantum mechanics explains why electrons exist in discrete energy levels around an atom's nucleus.

The transition of an electron from one orbit to another is a quantum event, all governed by the principles of quantum mechanics. Take the exercise given: the electron's movement from a higher to a lower energy level results in the emission of light. This process is known as 'quantum jump' and can be observed and measured via spectroscopic techniques.

Impact on Chemistry

The impact of quantum mechanics in chemistry is far-reaching. From the structure of the periodic table to the behavior of chemical bonds, quantum principles shape our understanding of matter at the most fundamental level. The calculation of the energy of an electron transitioning between orbits, for example, is based on a quantum mechanical understanding of energy quantization.

Bohr Model of the Atom

The Bohr model of the atom is a historical concept that revolutionized the way scientists viewed atomic structure. Despite its limitations for more complex atoms, the Bohr model is still a crucial teaching tool for explaining atomic behavior in simple systems like the hydrogen atom.

In the Bohr model, electrons orbit the nucleus in circular paths, with each orbit corresponding to a specific, allowable energy state. This model introduced the idea of quantization in atomic physics, with the transitions of an electron from one orbit to another being associated with the absorption or emission of a quantum of energy—in the form of light.

Applying the Bohr Model

Understanding the exercise's hydrogen electron transition can be approached using the Bohr model constructs. By calculating the energy difference between orbits n=5 and n=2, the model provides a visual framework to grasp how the energy of light is correlated to electron transitions. The process, as visualized by Bohr's concept, simplifies complex quantum mechanics into a more approachable illustration of atomic behavior. While modern quantum mechanics has provided a more complex and accurate portrayal of atomic structure, the essential principles introduced by Bohr's model remain a foundation for students exploring the world of atomic physics and chemistry.