Question

Electrons with the same value of quantum number \(n\) are said to occupy the same electron shell \(K, L, M, N,\) etc. Calculate the maximum allowed number of electrons for the a) \(K\) shell, b) \(L\) shell, and c) \(M\) shell.

Short answer

Answer: The maximum allowed number of electrons for the K, L, and M electron shells are 2, 8, and 18 respectively.

Step-by-step solution

Understand the formula

We are using the formula to calculate the maximum allowed number of electrons in a given shell. The formula is: \(max \, electrons \, = 2n^2\) where \(n\) represents the principal quantum number.

Calculate for the K shell

For the \(K\) shell, the principal quantum number \(n = 1\). Using the formula, we have: \(max \, electrons \, (K) = 2(1^2) = 2(1) = 2\). So, the maximum allowed number of electrons for the \(K\) shell is 2.

Calculate for the L shell

For the \(L\) shell, the principal quantum number \(n = 2\). Using the formula, we have: \(max \, electrons \, (L) = 2(2^2) = 2(4) = 8\). So, the maximum allowed number of electrons for the \(L\) shell is 8.

Calculate for the M shell

For the \(M\) shell, the principal quantum number \(n = 3\). Using the formula, we have: \(max \, electrons \, (M) = 2(3^2) = 2(9) = 18\). So, the maximum allowed number of electrons for the \(M\) shell is 18. In summary, the maximum allowed number of electrons for the: a) \(K\) shell is 2, b) \(L\) shell is 8, and c) \(M\) shell is 18.

Key concepts

Understanding Quantum Numbers

Quantum numbers are like addresses for electrons in an atom. They help us determine where an electron is likely to be found. There are four quantum numbers, but in this topic, we focus on the principal quantum number. The principal quantum number, represented by the symbol \( n \), tells us how far an electron is from the nucleus. It essentially defines the size of the electron shell or "energy level" where the electron resides.

• The larger the value of \( n \), the further the shell is from the nucleus and the higher the energy level.
• Values of \( n \) are positive integers like 1, 2, 3, and so on.
• Each integer corresponds to a specific electron shell: \( n=1 \) is the \( K \) shell, \( n=2 \) is the \( L \) shell, \( n=3 \) is the \( M \) shell, etc.

Understanding these numbers is crucial because they help us calculate how many electrons can fit in an electron shell. Let's explore this in the next parts.

The Role of Principal Quantum Number

The principal quantum number \( n \) is one of the four quantum numbers that define the unique state of each electron in an atom. It directly influences two key features of electron shells:

• Energy Level: Higher values of \( n \) mean higher energy levels. Electrons in these levels are further from the nucleus and are thus easier to remove.
• Shell Capacity: The bigger the \( n \), the more electrons the shell can accommodate. This capacity is calculated using the formula which we'll describe in the next section.

When \( n = 1 \), you're looking at the \( K \) shell, the closest shell to the nucleus, which holds the least energy and electrons. For \( n = 2 \), the \( L \) shell, a bit farther than \( K \), holds more electrons and more energy. With \( n = 3 \), the electron occupies the \( M \) shell, further increasing the number of electrons and energy.
This intuitive grasp of \( n \) enriches our understanding of atomic structure and predicts how atoms might interact or bond with one another.

Applying the Maximum Electrons Formula

Electrons in shells have a maximum limit, dictated by the formula \(2n^2\). This is the maximum electrons formula that scientists use to calculate the number of electrons that can fit in each electron shell.Using the formula:

• For \( n = 1 \) (\( K \) shell): \( \text{Max Electrons} = 2 \times 1^2 = 2 \). Therefore, the \( K \) shell can hold up to 2 electrons.
• For \( n = 2 \) (\( L \) shell): \( \text{Max Electrons} = 2 \times 2^2 = 8 \). Hence, the \( L \) shell has the capacity of 8 electrons.
• For \( n = 3 \) (\( M \) shell): \( \text{Max Electrons} = 2 \times 3^2 = 18 \). This means the \( M \) shell can accommodate up to 18 electrons.

This formula is pivotal because it helps in visualizing and understanding the population of electrons within atoms, and it's fundamental in explaining the chemical properties and reactivity of elements. The larger the shell, the more electrons it can handle, which impacts how atoms bond and the energy required for electron transitions.