Problem 56 - Chemistry: The Central Science

Question

How many possible values for \(l\) and \(m_{l}\) are there when (a) \(n=3\), (b) \(n=5\) ?

Step-by-Step Solution

Answer

For \(n=3\), there are 9 possible values of \(l\) and \(m_l\), and for \(n=5\), there are 25 possible values of \(l\) and \(m_l\).

Step 1: Find possible values of the angular momentum quantum number (l) for n=3

Since \(l\) ranges from 0 to \(n-1\), for \(n=3\), we have \(l=0,1,2\).

Step 2: Find possible values of the magnetic quantum number (\(m_l\)) for each value of l

For each value of \(l\), the magnetic quantum number (\(m_l\)) ranges from \(-l\) to \(l\). For \(l=0\), \(m_l=0\) (1 value) For \(l=1\), \(m_l=-1,0,1\) (3 values) For \(l=2\), \(m_l=-2,-1,0,1,2\) (5 values) In total, we have 1+3+5=9 possible values of \(l\) and \(m_l\) for \(n=3\). #b) n=5#

Step 1: Find possible values of the angular momentum quantum number (l) for n=5

Since \(l\) ranges from 0 to \(n-1\), for \(n=5\), we have \(l=0,1,2,3,4\).

Step 2: Find possible values of the magnetic quantum number (\(m_l\)) for each value of l

For each value of \(l\), the magnetic quantum number (\(m_l\)) ranges from \(-l\) to \(l\). For \(l=0\), \(m_l=0\) (1 value) For \(l=1\), \(m_l=-1,0,1\) (3 values) For \(l=2\), \(m_l=-2,-1,0,1,2\) (5 values) For \(l=3\), \(m_l=-3,-2,-1,0,1,2,3\) (7 values) For \(l=4\), \(m_l=-4,-3,-2,-1,0,1,2,3,4\) (9 values) In total, we have 1+3+5+7+9=25 possible values of \(l\) and \(m_l\) for \(n=5\).

Key Concepts

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \(l\), is a fundamental concept in quantum mechanics. It describes the shape of an electron's orbital and is integral in determining the electron's energy levels within an atom.

• The allowed values for \(l\) range from 0 to \(n-1\), where \(n\) is the principal quantum number.
• If the principal quantum number \(n=3\), \(l\) can take on the values 0, 1, and 2. These correspond to s, p, and d orbitals, respectively.
• As \(n\) increases, more intricate shapes and orbitals become available because \(l\) can take on more values.

The possibilities for \(l\) help elucidate the complex arrangements of electrons in atoms, contributing to our understanding of chemical bonding and atomic structure.

Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_l\), further determines the orientation of an electron's orbital in space. It builds upon the angular momentum quantum number to provide a more complete picture of an electron's behavior.

• For a given \(l\), \(m_l\) can take values ranging from \(-l\) to \(+l\).
• This means that each kind of orbital defined by \(l\) can have multiple orientations in a three-dimensional space. For example, if \(l=1\), then \(m_l\) could be -1, 0, or 1, showing three distinct orientations.
• The number of possible \(m_l\) values influences how electrons fill up orbitals and thus impacts an atom's electron configuration.

Understanding \(m_l\) allows chemists and physicists to predict the likelihood of an electron's placement in various energy states and to infer an element's magnetic properties.

Quantum Mechanics

Quantum mechanics is the overarching framework that defines the behavior of particles on very small scales, such as electrons in an atom. Within this context, quantum numbers, including the angular momentum quantum number \(l\) and the magnetic quantum number \(m_l\), play a critical role.

• Quantum mechanics explains that particles like electrons behave both as particles and waves, a core concept represented through wave functions.
• The Schrödinger equation is a fundamental equation in quantum mechanics that provides these wave functions, describing how the quantum state of a physical system changes over time.
• Quantum numbers arise from solutions to the Schrödinger equation, providing discrete values that define the quantum state of an electron.

Quantum mechanics challenges classical physics by providing probabilistic predictions, rather than deterministic outcomes, which are crucial for explaining phenomena like electron configurations and chemical bonds.