Question

What is the largest number of electrons with the same pair of values for \(n\) and \(\ell\) that an atom can have?

Short answer

Answer: The largest number of electrons with the same pair of values for n and l within an atom is given by the expression 4n - 2, where n represents the principal quantum number.

Step-by-step solution

Understanding the quantum numbers and the Pauli Exclusion Principle

The principal quantum number (n) determines the energy level of an electron, ranging from 1 to \(\infty\). The azimuthal quantum number (l) determines the shape of the electron orbital and ranges from 0 to n-1. For a given pair of values of n and l, the magnetic quantum number (m_l) ranges from -l to +l. Lastly, the electron spin quantum number (m_s) can have two possible values: +1/2 or -1/2. The Pauli Exclusion Principle states that no two electrons within an atom can have the same set of quantum numbers (n, l, m_l, and m_s).

Identify the maximum number of electrons with the same pair of values for n and l

Given any pair of values for n and l, there can be a range of values for the magnetic quantum number (m_l) from -l to +l, resulting in a total of 2l+1 possible different values of m_l. Since each of these m_l values can be associated with an electron having m_s either +1/2 or -1/2 (two possible values), the maximum number of electrons with the same pair of n and l values would be: Number of electrons = (2l + 1) x 2 Let's simplify the expression: Number of electrons = 4l + 2

Find the largest possible value for l within an atom

In an atom, the largest possible value for l arises when it equals n-1 (for any given n). For example, in the ground state (n=1), there could be one possible l value (l=0), but for higher n values, the number of possible l values increases. Therefore, the largest number of electrons with the same pair of n and l values can be obtained when l is maximized to (n-1).

Calculate the maximum number of electrons with the same pair of values for n and l

Using the equation derived in Step 2 and plugging in the maximum value of l (n-1): Number of electrons = 4(n-1) + 2 Simplify the expression: Number of electrons = 4n - 4 + 2 Number of electrons = 4n - 2 Thus, the maximum possible number of electrons with the same pair of values for n and l within an atom is given by the expression 4n - 2, where n represents the principal quantum number.