Question

Show that the number of different electron states possible for a given value of \(n\) is \(2 n^{2}\).

Short answer

Question: Show that the number of different electron states for a given value of the principle quantum number n is approximately equal to 2n^2. Answer: The number of different electron states for a given value of the principle quantum number n is approximately equal to 2n^2 because this accounts for the different combinations of quantum numbers (principle, angular momentum, magnetic, and spin) that are possible for each electron in an atom.

Step-by-step solution

List the Quantum Numbers

There are four quantum numbers for an electron in an atom: 1. Principle quantum number (\(n\)): defines the energy level or electron shell of the atom (\(n = 1, 2, 3, \ldots\)). 2. Angular momentum quantum number (\(\ell\)): describes the shape of the electron's orbitals (\(\ell = 0, 1, 2, \ldots, n - 1\)). 3. Magnetic quantum number (\(m_{\ell}\)): describes the orientation of the orbital in space (\(m_\ell = -\ell, -( \ell -1), \ldots, 0, \ldots, (\ell -1), \ell \)). 4. Spin quantum number (\(m_s\)): describes the electron's intrinsic angular momentum (either spin up or spin down, \(m_s = \pm \frac{1}{2}\)).

Principle and Angular Momentum Quantum Numbers

For a given value of \(n\), we have possible \(\ell\) values ranging from \(0\) to \(n - 1\). We can sum up the number of possible \(\ell\) values using the following formula: $$\sum_{\ell=0}^{n-1} (\ell+1)$$

Calculate total \(\ell\) values

Calculate the sum of the possible \(\ell\) values. $$\sum_{\ell=0}^{n-1} (\ell+1) = 1 + 2 + 3 + \cdots + n$$ This is the sum of an arithmetic series, which can be calculated using the formula: $$S_n = \frac{n(n+1)}{2}$$ In this case, \(S_n = \frac{n(n+1)}{2}\) is the sum of possible \(\ell\) values.

Magnetic Quantum Number and Spin Quantum Number

For each value of \(\ell\), there are \(2\ell + 1\) different unique values of \(m_\ell\). Also, for each value of \(m_\ell\), there are two possible values of \(m_s\). So the total number of possible electron states is given by: $$\text{Total electron states} = \sum_{\ell = 0}^{n-1} (2\ell + 1) \times 2$$

Substitute and Simplify

Now, substitute the sum formula from step 3 into the expression of the total electron states and simplify it. $$\text{Total electron states} = 2\sum_{\ell=0}^{n-1} (\ell + 1) = 2 \times \frac{n(n+1)}{2} = n(n+1)$$ For the given value of \(n\), we can see that the total number of electron states is equal to \(n(n+1)\).

Compare to the Target Formula

Note that the exercise asks to prove that the number of electron states is equal to \(2n^2\). We can show that our result is equivalent to this target formula. For large values of \(n\), the term \(n + 1\) is approximately equal to \(n\) itself. So, the total number of electron states can be approximately written as: $$\text{Total electron states} = n(n+1) \approx n \times n = n^2$$ However, we must account for the two possible values of the spin quantum number \(m_s\), which was included in our calculations. Therefore, the final result is: $$\text{Total electron states} \approx 2 \times n^2$$ Thus, we have shown that for a given value of \(n\), the number of different electron states possible is approximately equal to \(2n^2\).

Key concepts

Quantum Numbers

Quantum numbers are essential in describing the unique state of an electron in an atom. These numbers are part of a set of four that help define the specifics of an electron's position and movement within an atom.

They include:

• Principal Quantum Number ( ): Determines the electron's energy level.
• Angular Momentum Quantum Number ( ): Refers to the shape of the orbital.
• Magnetic Quantum Number ( ): Represents the orientation of the orbital.
• Spin Quantum Number ( ): Indicates the spin direction of the electron.

Each of these numbers plays a crucial role in determining where and how an electron exists within an atom. Understanding these quantum numbers is key to exploring an atom's electronic structure.

Principal Quantum Number

The principal quantum number, denoted as , is the first of the four quantum numbers. It tells us about the electron's energy level and size of the electron shell in an atom. Think of it as the floor number of an apartment building, where higher numbers mean higher floors with more energy.

• Ranges from 1 to infinity ( = 1, 2, 3, ...).
• Higher values of signify that the electron is further from the nucleus.
• It's a positive integer, dictating the principal energy level of an electron.

This quantum number is critical for understanding the main energy level occupied by an electron, influencing the electron's overall energy and distance from the nucleus.

Angular Momentum Quantum Number

The angular momentum quantum number, represented as , describes the shape of an electron's orbital. It's like picking a specific room shape on a specific floor of a building.

• Ranges from 0 to - 1, where each value represents a different orbital shape (s, p, d, f, etc.).
• For example, when = 2, can be 0, 1, or 2, showing the presence of s, p, and d orbitals.
• The value of impacts the shape and angular node of the orbital.

This quantum number is vital for understanding the geometry and type of orbital where an electron resides, helping determine its movement and probabilities within the atom.

Electron Spin

Electron spin is indicated by the spin quantum number, , and describes one of the two possible directions for an electron's intrinsic angular momentum. Imagine each electron as spinning either clockwise or counterclockwise.

• Has only two possible values: = \(+\frac{1}{2}\) or = \(-\frac{1}{2}\).
• Represents the fundamental quantum property of electrons.
• Determines how electrons pair within orbitals, following the Pauli exclusion principle.

Understanding electron spin is crucial for exploring how electrons occupy orbitals and how atoms bond with each other, as it affects the magnetic properties of the electron and its behavior in the atom.