Problem 7 - An Introduction to Physical Science

Question

What is the energy in eV of the electron of a hydrogen atom for the orbit designated \(n=2 ?\)

Step-by-Step Solution

Answer

The energy is -3.4 eV.

Step 1: Understand the Formula

For a hydrogen atom, the energy of an electron in the nth orbit is given by the formula: \[ E_n = -13.6 \, \text{eV} \frac{1}{n^2} \] where \(E_n\) is the energy of the electron, and \(n\) is the principal quantum number of the orbit.

Step 2: Substitute the Value of n

Substitute \( n = 2 \) into the formula to find the energy for this orbit: \[ E_2 = -13.6 \, \text{eV} \frac{1}{2^2} \] which simplifies to: \[ E_2 = -13.6 \, \text{eV} \frac{1}{4} \]

Step 3: Compute the Energy Value

Calculate the energy value: \[ E_2 = -13.6 \, \text{eV} \times \frac{1}{4} = -3.4 \, \text{eV} \] So, the energy of the electron in the \( n=2 \) orbit is \( -3.4 \) eV.

Key Concepts

Electron Energy in Orbits

The concept of electron energy in orbits is a fundamental aspect of quantum mechanics, especially when dealing with hydrogen atoms. Atoms have electrons that move around the nucleus in specific paths called orbits. Each orbit corresponds to an energy level, where electrons can reside. In a hydrogen atom, the energy of an electron in a particular orbit is negative, signifying that it is bound to the nucleus. The specific energy value indicates how strongly the electron is bound. A more negative energy means a stronger attraction between the electron and the nucleus. For each orbit an electron can occupy, there is a discrete energy value. Calculating the energy of an electron in a specific orbit helps us understand atomic behavior and spectral lines observed in experiments. The electron energy levels in an atom are crucial for predicting the atom's reactions and interactions with electromagnetic radiation.

Principal Quantum Number

The principal quantum number, often denoted by the symbol \( n \), is a critical component of quantum mechanics that helps describe the state of an electron in an atom. It is an integer value that indicates the relative size and energy of atomic orbitals. Higher values of \( n \) mean the electron is in an orbital further from the nucleus, associated with higher potential energy.Key aspects of the principal quantum number:

• It determines the size of the orbital: Larger \( n \) values correspond to larger orbitals.
• It affects the energy of the electron: Higher \( n \) means higher energy levels.
• It defines the shell in which an electron resides: For example, \( n=1 \) is the first shell, \( n=2 \) is the second shell, and so on.

The principal quantum number gives us a simple way to understand and calculate electron configurations and predict how they are distributed around the nucleus.

Electronvolt Calculation

Calculating the energy of an electron in terms of electronvolts (eV) is essential for understanding atomic energy levels. An electronvolt is a unit of energy often used in atomic and nuclear physics. It represents the energy gained or lost by an electron when it moves across an electric potential difference of one volt.The formula for calculating electron energy in an orbit is:\[ E_n = -13.6 \, \text{eV} \frac{1}{n^2} \]where:

• \( E_n \) is the energy of the electron in orbit \( n \).
• \(-13.6 \, \text{eV}\) is the ground state energy of a hydrogen atom.
• \( n \) is the principal quantum number.

Substituting the appropriate \( n \) value allows us to find the energy level for specific orbits quickly. For instance, substituting \( n = 2 \) yields \(-3.4 \, \text{eV}\). This calculation helps students understand how energy levels vary with different electron orbits. Understanding these values also plays a role in analyzing spectral lines and predicting the behavior of atoms under different conditions.