Let's analyze the image and solve the problem step by step.
---
Image Description:
The image shows a diagram of
5f orbitals with electrons filled according to quantum numbers. The title says:
>
"S P D F Orbitals - Quantum #s"
Below that, we see:
- A row labeled `5f¹³`, meaning the 5f subshell has 13 electrons.
- Below this, there are
7 boxes (representing the 7 f-orbitals), each corresponding to a magnetic quantum number $ m_l $ from $-3$ to $+3$:
$$
m_l = -3,\ -2,\ -1,\ 0,\ +1,\ +2,\ +3
$$
- Each box has up/down arrows representing electron spins.
- Electrons are filled following
Hund’s Rule: maximize parallel spins before pairing.
- One electron in the $ m_l = +2 $ orbital is
circled, and its spin is
down ($ \downarrow $).
We're asked to determine the
quantum numbers for the
last electron added (the circled one), which is the 13th electron in the 5f subshell.
---
Step-by-step Analysis:
####
1. Principal Quantum Number (n):
The electron is in the
5f orbital →
$$
\boxed{n = 5}
$$
####
2. Azimuthal Quantum Number (ℓ):
For an
f orbital, $ \ell = 3 $.
$$
\boxed{\ell = 3}
$$
####
3. Magnetic Quantum Number (mₗ):
The circled electron is in the $ m_l = +2 $ orbital.
$$
\boxed{m_l = +2}
$$
####
4. Spin Quantum Number (mₛ):
The arrow is
down, which means:
$$
\boxed{m_s = -\frac{1}{2}}
$$
---
✔ Final Answer:
The quantum numbers for the
last (13th) electron in the $ 5f^{13} $ configuration are:
$$
\boxed{
n = 5,\quad
\ell = 3,\quad
m_l = +2,\quad
m_s = -\frac{1}{2}
}
$$
---
🔍 Explanation:
- The 5f subshell has 7 orbitals ($ m_l = -3 $ to $ +3 $).
- Each orbital holds 2 electrons (with opposite spins).
- Total capacity: $ 7 \times 2 = 14 $ electrons.
- Here, only 13 electrons are present →
one unpaired electron.
- Electrons fill all orbitals with
parallel spins first (Hund’s Rule).
- So, first 7 electrons go into each orbital with $ \uparrow $.
- Then, the next 6 electrons pair up as $ \downarrow $ in the first 6 orbitals.
- The
13th electron is the
second electron in the $ m_l = +2 $ orbital → it pairs with the first one.
- Since it's paired, it must have
opposite spin, so $ m_s = -\frac{1}{2} $.
- It occupies $ m_l = +2 $, so that's its magnetic quantum number.
---
📌 Summary:
This is a classic example of determining quantum numbers from an electron configuration using orbital diagrams. The key is identifying:
- Which subshell (gives $ n $ and $ \ell $),
- Which orbital (gives $ m_l $),
- And the spin direction (gives $ m_s $).
✔ All four quantum numbers are correctly determined.
Parent Tip: Review the logic above to help your child master the concept of quantum numbers worksheets.
