Question

What is the formula for the maximum number of electrons in each shell of the Bohr atom? How many electrons are allowed in the \(n=2\) shell? The \(n=6\) shell?

Short answer

The formula for the maximum number of electrons in each shell of the Bohr atom is \(2n^2\), where 'n' is the principal quantum number of the shell. For the n=2 shell, there are 8 electrons allowed, and for the n=6 shell, there are 72 electrons allowed.

Step-by-step solution

Understanding the Bohr atom and electron shells

The Bohr model for atoms describes electrons moving around the nucleus in specific energy levels, called shells or orbits. The energy levels correspond to different distances from the nucleus. Each shell can hold a specific maximum number of electrons.

Find the formula for the maximum number of electrons in each shell

In a Bohr atom, each shell can hold a maximum number of electrons represented by the formula: \(2n^2\) Where 'n' is the principal quantum number of the shell, and it can have positive integer values (1, 2, 3, ...). This formula gives us the maximum number of electrons that can be accommodated in each shell.

Calculate the number of electrons in n=2 shell

Using the formula from Step 2, let's find the number of electrons allowed in the n=2 shell. n = 2 Maximum electrons = \(2n^2\) Maximum electrons in n=2 shell = \(2(2^2)\) = \(2(4)\) = 8 There are 8 electrons allowed in the n=2 shell.

Calculate the number of electrons in n=6 shell

Now let's find the number of electrons allowed in the n=6 shell. n = 6 Maximum electrons = \(2n^2\) Maximum electrons in n=6 shell = \(2(6^2)\) = \(2(36)\) = 72 There are 72 electrons allowed in the n=6 shell.

Key concepts

Electron Shells

The concept of electron shells is fundamental to understanding how atoms are structured and how they behave in different chemical reactions. Electron shells, also known as energy levels, are regions surrounding the nucleus of an atom where electrons are most likely to be found.

Each shell is characterized by its own unique energy level and is commonly denoted by the principal quantum number, represented as 'n'. The value of 'n' starts at 1 for the shell closest to the nucleus and increases as you move further away. Electrons naturally fill the lowest available energy levels first before moving to higher levels, a principle known as the Aufbau principle.

Since each shell can hold a certain maximum number of electrons given by the formula 2n², the shells are filled sequentially. For instance, the first shell (n=1) can hold up to 2 electrons, the second shell (n=2) up to 8, and so on. This pattern is crucial for accurately predicting the electronic configuration of elements and their subsequent chemical properties.

Principal Quantum Number

The principal quantum number, symbolized as 'n', is one of four quantum numbers that describe the unique quantum state of an electron in an atom. It is a positive integer and it determines the electron shell, or energy level, in which the electron resides. The value of 'n' defines not only the shell but also the size of the electron cloud and the energy associated with the electrons in that shell.

As 'n' increases, the energy level and the average distance of the electrons from the nucleus also increase, meaning electrons in higher shells are less tightly bound to the nucleus compared to those in lower shells. The principal quantum number typically ranges from 1, for the innermost shell, and increases outward. In terms of capacity, the related formula, 2n², illustrates that the potential to accommodate electrons within a shell expands dramatically with higher values of 'n'.

Understanding the principal quantum number is essential when deducing an atom's electron configuration and predicting the types of chemical bonds an atom may form based on its electron arrangement.

Atomic Structure

Atomic structure refers to the organization of protons, neutrons, and electrons within an atom. At the center is the nucleus, comprising protons and neutrons, which account for most of the atom's mass. Surrounding the nucleus are the electron shells, where electrons orbit in various energy levels.

The atomic structure is determined mainly by the interactions between these subatomic particles and quantum mechanics principles. For example, electrons occupy discrete energy levels or shells, and the specific arrangement of these electrons around the nucleus — as governed by quantum numbers and the Pauli exclusion principle — greatly influences an atom's stability and reactivity.

When we consider the Bohr model, which simplifies atomic structure into a solar system-like layout, it provides a basic yet effective understanding of how electrons are arranged and how these arrangements impact an atom's properties. However, modern atomic theory adopts the quantum mechanical model for a more accurate and complex depiction, where electron positions are described as probabilities rather than fixed orbits.