Problem 40 - Chemistry The Central Science In Si Units

Question

Consider a transition of the electron in the hydrogen atom from \(n=8\) to \(n=3\). (a) Is \(\Delta E\) for this process positive or negative? (b) Determine the wavelength of light that is associated with this transition. Will the light be absorbed or emitted? (c) In which portion of the electromagnetic spectrum is the light in part (b)?

Step-by-Step Solution

Answer

(a) \( \Delta E \) is negative, indicating light is emitted. (b) Calculate wavelength to find it's near 656 nm, indicating visible light emission (red portion). (c) This light is in the visible spectrum.

Step 1: Understanding Energy Change

In quantum mechanics, when an electron transitions to a lower energy level, energy is released. For a hydrogen atom, the energy of an electron at a given energy level \( n \) is given by: \[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]For a transition from \( n=8 \) to \( n=3 \), we calculate \( \Delta E = E_3 - E_8 \).Thus, \( \Delta E = -\frac{13.6}{3^2} + \frac{13.6}{8^2} \).Since the electron moves to a lower energy (\( n=3 \)), \( \Delta E \) will be negative, indicating energy emission.

Step 2: Calculating the Wavelength

Using the energy change \( \Delta E \) calculated:\[ \Delta E = -13.6 \left ( \frac{1}{3^2} - \frac{1}{8^2} \right ) \text{ eV} \]Convert \( \Delta E \) from eV to Joules (1 eV = \(1.602 \times 10^{-19}\) J).Then, use the formula: \[ \lambda = \frac{hc}{\Delta E} \]where \( h = 6.626 \times 10^{-34} \text{ J s} \) and \( c = 3.00 \times 10^8 \text{ m/s} \). Calculate \( \lambda \) in meters.

Step 3: Determining Emission or Absorption

An energy release (negative \( \Delta E \)) means light is emitted as the electron moves to a lower energy level.

Step 4: Identifying the Electromagnetic Spectrum

After calculating the wavelength \( \lambda \), compare it to known ranges of the electromagnetic spectrum. If \( \lambda \) is within visible light (approximately 400 nm to 700 nm), infra-red, or ultraviolet ranges, classify it appropriately based on its magnitude.

Key Concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. In this realm, particles like electrons do not follow the classical rules we're used to seeing in everyday life. Instead, they exhibit unique behavior that can be described using probabilities. One fundamental aspect of quantum mechanics is quantization, where certain properties like energy are only found in discrete set values, or "quanta."

For a hydrogen atom, this quantization ensures that electrons occupy specific energy levels. These levels are not continuous, but rather distinct. When an electron moves between these levels, it must gain or lose precise amounts of energy, often in the form of light. This process is pivotal in understanding phenomena like the emission or absorption of light by atoms.

Electron Energy Levels

In the realm of quantum mechanics, electron energy levels are distinct states that an electron can occupy within an atom. Each level corresponds to a specific amount of energy that the electron around the nucleus can have. These levels are indicated by the principal quantum number, often denoted as "n."

For example, in a hydrogen atom, the energy levels are given by the formula:

• \( E_n = -\frac{13.6 \, \text{eV}}{n^2} \)

This negative sign indicates that energy is required to remove the electron from the atom. The higher the "n," the further away the electron is from the nucleus, and the more energy it has. When an electron moves from a higher energy level, such as \( n=8 \), to a lower one, like \( n=3 \), it releases energy in the form of electromagnetic radiation. This release results in a negative energy change \( (\Delta E) \), signifying that energy leaves the system.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation. Radiation can be characterized by its wavelength or frequency, ranging from long-wavelength radio waves to short-wavelength gamma rays. Within this spectrum, visible light is just a small portion, spanning wavelengths from approximately 400 nm to 700 nm. Each type of radiation within the spectrum carries energy, and its role depends on its precise wavelength.

When an electron transitions between energy levels, the emitted or absorbed light falls somewhere on this spectrum. Knowing which portion the light belongs to allows us to understand its properties and potential effects. For instance, light with wavelengths shorter than visible light, like ultraviolet, carries more energy and can impact living cells. In our specific case of the hydrogen atom transition, identifying the resulting wavelength ensures we understand where it resides within the electromagnetic spectrum.

Light Emission

Light emission occurs when an electron in an atom transitions from a higher energy level to a lower one, releasing energy as electromagnetic radiation. This can be visualized as the electron "falling" closer to the nucleus, resulting in a loss of energy which is emitted as light. The difference between the initial and final energy levels determines the energy of the emitted light.

Planck's constant \( h \) and the speed of light \( c \) are crucial in calculating the wavelength \( \lambda \) of emitted light using the formula:

• \( \lambda = \frac{hc}{\Delta E} \)

This formula shows that the wavelength is inversely proportional to the energy change \( \Delta E \). Thus, larger energy transitions result in shorter wavelengths. By calculating \( \lambda \), we can specify the type of light emitted and, subsequently, locate it within the electromagnetic spectrum. Emission plays a vital role in deciphering fundamental properties of atoms and molecules, as it provides insight into electron dynamics and forces within an atom.