Where:

• $E_{\text{photon}}$ = energy of incoming photon (constant)
• $IE$ = ionization energy (binding energy of electron)
• $KE$ = kinetic energy of ejected electron

Rearranged:

$IE = E_{\text{photon}} - KE$

PES Spectrum Features

X-axis: Binding Energy (Ionization Energy)

• Units: MJ/mol or eV
• Direction: Usually increases left to right
• Higher binding energy = electron held more tightly
• Represents energy needed to remove electron

Y-axis: Relative Number of Electrons

• Also called: Intensity, Signal, or Counts
• Height of peak indicates number of electrons with that binding energy
• Area under peak is proportional to number of electrons

Peaks

Each peak represents electrons from a specific orbital/sublevel:

• Position (x): Binding energy (how tightly held)
• Height (y): Number of electrons in that orbital

Example: A peak at 1.4 MJ/mol with height of 2 means 2 electrons have binding energy of 1.4 MJ/mol

Reading PES Data

General Trends

Binding energy increases with:

1. Closer to nucleus (lower n value)
2. Higher nuclear charge (more protons)
3. Less shielding (inner vs. outer electrons)

Typical order (increasing binding energy):

$\text{Valence orbitals} < \text{Inner orbitals} < \text{Core orbitals}$

For multi-electron atoms:

$ns < np < (n-1)d < (n-1)p < (n-1)s < ... < 1s$

Interpreting Peaks

Example: Nitrogen (N)

Electron configuration: $1s^2 2s^2 2p^3$

Expected PES spectrum:

1. Peak at ~1.4 MJ/mol, height = 3: Three 2p electrons (valence)
2. Peak at ~1.8 MJ/mol, height = 2: Two 2s electrons (valence)
3. Peak at ~10 MJ/mol, height = 2: Two 1s electrons (core)

Three peaks total (three different orbitals)

Information from PES

1. Number of Occupied Orbitals

Number of peaks = Number of occupied sublevels

• 3 peaks → 3 occupied sublevels (e.g., 1s, 2s, 2p)
• 5 peaks → 5 occupied sublevels (e.g., 1s, 2s, 2p, 3s, 3p)

2. Number of Electrons in Each Orbital

Height/area of peak = Number of electrons

• Peak height of 2 → 2 electrons
• Peak height of 6 → 6 electrons

3. Relative Energies

Peak position reveals how tightly electrons are held:

• Rightmost peak (highest binding energy) = 1s (closest to nucleus)
• Leftmost peak (lowest binding energy) = valence electrons (farthest from nucleus)

4. Identifying Elements

Combine information to write electron configuration, then identify element:

Example:

• 4 peaks with heights 2, 2, 6, 2 (left to right)
• Configuration: $1s^2 2s^2 2p^6 3s^2$
• Element: Magnesium (Mg)

Comparing PES Spectra

Same Period (e.g., C, N, O)

Trend: Binding energies increase across period

• More protons → stronger nuclear charge
• Same shielding (same inner electrons)
• Electrons held more tightly

Example:

• 1s electrons: $BE_C < BE_N < BE_O$
• More protons in O pull electrons tighter

Same Group (e.g., Li, Na, K)

Trend: Binding energies decrease down group

• More electron shells → more shielding
• Valence electrons farther from nucleus
• Weaker attraction despite more protons

Example:

• 2s in Li > 3s in Na > 4s in K

Core vs. Valence

Core electrons (inner shells):

• Much higher binding energy
• Closer to nucleus
• Less shielding

Valence electrons (outer shell):

• Lower binding energy
• Farther from nucleus
• More shielding

Large gap between core and valence peaks

Common PES Questions

Type 1: Write electron configuration from spectrum

Given: Peak positions and heights Find: Electron configuration and element

Steps:

1. Count peaks (number of sublevels)
2. Note heights (electrons per sublevel)
3. Assign orbitals from right to left (high to low BE)
4. Write configuration and identify element

Type 2: Predict spectrum from element

Given: Element or electron configuration Find: Number of peaks, relative heights, relative positions

Steps:

1. Write electron configuration
2. Count occupied sublevels (number of peaks)
3. Count electrons per sublevel (peak heights)
4. Order by binding energy (1s highest, valence lowest)

Type 3: Compare spectra

Given: Two elements Find: Differences in binding energies

Steps:

1. Consider nuclear charge (more protons → higher BE)
2. Consider shielding (more shells → lower BE)
3. Compare same orbitals between elements

Example: Sodium (Na)

Electron configuration: $1s^2 2s^2 2p^6 3s^1$

PES spectrum:

• Peak 1: 3s¹ (height = 1, lowest BE ~0.5 MJ/mol) - valence
• Peak 2: 2p⁶ (height = 6, medium BE ~3 MJ/mol) - inner
• Peak 3: 2s² (height = 2, high BE ~6 MJ/mol) - inner
• Peak 4: 1s² (height = 2, highest BE ~100 MJ/mol) - core

Total: 4 peaks

Note: Large gap between 3s and 2p (valence vs. core)

Key Relationships

Binding Energy and Atomic Number

Same orbital, different elements:

Higher atomic number (Z) → Higher binding energy

Example: 1s electrons

• Li (Z=3): ~6 MJ/mol
• C (Z=6): ~11 MJ/mol
• F (Z=9): ~80 MJ/mol

Sublevels in Same Shell

Energy order (increasing BE):

$ns < np < (n-1)d$

Example for n=3: $3s < 3p < 2d$

However, 3d is usually empty or partially filled, and 2d doesn't exist.

More common: $3s < 3p < 2p < 2s < 1s$

Applications

1. Elemental analysis: Identify unknown elements
2. Electron configuration confirmation: Verify predicted configurations
3. Bonding studies: Understand how electrons participate in bonds
4. Material characterization: Analyze surface composition
5. Support for quantum theory: Experimental evidence for orbital model

Tips for AP Exam

1. Right to left: Highest BE (rightmost) is always 1s
2. Count electrons: Sum of peak heights = total electrons = atomic number
3. Valence electrons: Leftmost peak(s) with lowest BE
4. Core electrons: Large gap separates from valence
5. Period = shells: Number of principal energy levels (n values)
6. Peaks ≠ Electrons: Number of peaks is number of sublevels, not total electrons

Step 2: Determine total electrons

Total electrons = 1 + 2 + 2 = 5 electrons

Step 3: Assign orbitals

From right to left (high to low BE):

• Highest BE (50 MJ/mol): 1s² (2 electrons, closest to nucleus)
• Medium BE (5 MJ/mol): 2s² (2 electrons, next shell)
• Lowest BE (0.5 MJ/mol): 2p¹ (1 electron, valence)

Step 4: Write electron configuration

$1s^2 2s^2 2p^1$

Step 5: Identify element

5 total electrons → Atomic number = 5 → Boron (B)

Answer: Element is Boron (B) with configuration $1s^2 2s^2 2p^1$

Verification:

• 3 peaks for 3 sublevels (1s, 2s, 2p) ✓
• Heights match electron counts ✓
• Total electrons = 5 ✓
• Binding energy order (1s > 2s > 2p) ✓