Atomic Structure — Hard Practice Quiz

A Chemistry cheat sheet for Atomic Structure — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Wave Equation: Speed of light equals wavelength times frequency.

$$c = \lambda \nu$$

Where: c = 3.00 × 10<sup>8</sup> m/s, λ = wavelength (m), ν = frequency (s<sup>-1</sup>)

Photon Energy: Energy of a photon.

$$E = h\nu$$

Where: h = Planck's constant (6.626 × 10<sup>-34</sup> J·s), ν = frequency (s<sup>-1</sup>)

de Broglie Wavelength: Matter as a wave.

$$\lambda = \frac{h}{mv}$$

Where: h = Planck's constant (6.626 × 10<sup>-34</sup> J·s), m = mass (kg), v = velocity (m/s)

Bohr Model Energy Levels: Energy of electron in hydrogen atom (Equation 6.4).

$$E_n = -\frac{2.18 \times 10^{-18} \text{ J}}{n^2}$$

Where: n = principal quantum number (1, 2, 3, ...)

Energy Change in Hydrogen: Energy change when electron transitions between levels (Equation 6.5).

$$\Delta E = E_{\text{final}} - E_{\text{initial}} = 2.18 \times 10^{-18} \text{ J} \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)$$

Where: n<sub>i</sub> = initial level, n<sub>f</sub> = final level

Heisenberg Uncertainty Principle: The uncertainty in position and momentum cannot both be zero.

$$\Delta x \times \Delta(mv) \geq \frac{h}{4\pi}$$

Where: Δx = uncertainty in position, Δ(mv) = uncertainty in momentum, h = Planck's constant

Effective Nuclear Charge: The net nuclear charge experienced by an electron (Equation 7.1).

$$Z_{\text{eff}} = Z - S$$

Where: Z = atomic number (number of protons), S = screening constant (number of core electrons)

<table style="width:100%; border-collapse: collapse; font-size: 0.85em;"> <tr style="border-bottom: 1px solid #ccc;"><th>n</th><th>l</th><th>Subshell</th><th>m<sub>l</sub> values</th><th># Orbitals</th></tr> <tr><td>1</td><td>0</td><td>1s</td><td>0</td><td>1</td></tr> <tr><td rowspan="2">2</td><td>0</td><td>2s</td><td>0</td><td>1</td></tr> <tr><td>1</td><td>2p</td><td>-1, 0, +1</td><td>3</td></tr> <tr><td rowspan="3">3</td><td>0</td><td>3s</td><td>0</td><td>1</td></tr> <tr><td>1</td><td>3p</td><td>-1, 0, +1</td><td>3</td></tr> <tr><td>2</td><td>3d</td><td>-2, -1, 0, +1, +2</td><td>5</td></tr> <tr><td rowspan="4">4</td><td>0</td><td>4s</td><td>0</td><td>1</td></tr> <tr><td>1</td><td>4p</td><td>-1, 0, +1</td><td>3</td></tr> <tr><td>2</td><td>4d</td><td>-2, -1, 0, +1, +2</td><td>5</td></tr> <tr><td>3</td><td>4f</td><td>-3 to +3</td><td>7</td></tr> </table>

$$\text{Quantum Numbers}$$

<table style="width:100%; border-collapse: collapse; font-size: 0.85em;"> <tr style="border-bottom: 1px solid #ccc;"><th>Element</th><th>Electrons</th><th>Configuration</th></tr> <tr><td>Li</td><td>3</td><td>1s<sup>2</sup> 2s<sup>1</sup></td></tr> <tr><td>Be</td><td>4</td><td>1s<sup>2</sup> 2s<sup>2</sup></td></tr> <tr><td>B</td><td>5</td><td>1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>1</sup></td></tr> <tr><td>C</td><td>6</td><td>1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>2</sup></td></tr> <tr><td>N</td><td>7</td><td>1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>3</sup></td></tr> <tr><td>Ne</td><td>10</td><td>1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup></td></tr> <tr><td>Na</td><td>11</td><td>1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> 3s<sup>1</sup></td></tr> </table>

$$\text{Electron Configurations (Light Elements)}$$

<table style="width:100%; border-collapse: collapse; font-size: 0.85em;"> <tr style="border-bottom: 1px solid #ccc;"><th colspan="2">Group 2A</th><th colspan="2">Group 3A</th></tr> <tr><td>Be</td><td>[He] 2s<sup>2</sup></td><td>B</td><td>[He] 2s<sup>2</sup> 2p<sup>1</sup></td></tr> <tr><td>Mg</td><td>[Ne] 3s<sup>2</sup></td><td>Al</td><td>[Ne] 3s<sup>2</sup> 3p<sup>1</sup></td></tr> <tr><td>Ca</td><td>[Ar] 4s<sup>2</sup></td><td>Ga</td><td>[Ar] 3d<sup>10</sup> 4s<sup>2</sup> 4p<sup>1</sup></td></tr> <tr><td>Sr</td><td>[Kr] 5s<sup>2</sup></td><td>In</td><td>[Kr] 4d<sup>10</sup> 5s<sup>2</sup> 5p<sup>1</sup></td></tr> <tr><td>Ba</td><td>[Xe] 6s<sup>2</sup></td><td>Tl</td><td>[Xe] 4f<sup>14</sup> 5d<sup>10</sup> 6s<sup>2</sup> 6p<sup>1</sup></td></tr> </table>

$$\text{Group Electron Configurations}$$

Practice quiz

  1. A photon is emitted from a hydrogen atom when an electron transitions from $n=4$ to $n=2$. What is the wavelength of this emitted photon? Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$ and $c = 3.00 \times 10^8 \text{ m/s}$.

    • $486 \text{ nm}$
    • $656 \text{ nm}$
    • $434 \text{ nm}$
    • $410 \text{ nm}$

    Answer: $486 \text{ nm}$

  2. An electron is accelerated to a velocity such that its de Broglie wavelength is $1.00 \times 10^{-10} \text{ m}$. If the uncertainty in its position is $1.00 \text{ pm}$ ($1.00 \times 10^{-12} \text{ m}$), what is the minimum uncertainty in its velocity? Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$ and electron mass $m_e = 9.109 \times 10^{-31} \text{ kg}$.

    • $5.79 \times 10^7 \text{ m/s}$
    • $1.16 \times 10^8 \text{ m/s}$
    • $2.90 \times 10^7 \text{ m/s}$
    • $1.00 \times 10^6 \text{ m/s}$

    Answer: $5.79 \times 10^7 \text{ m/s}$

  3. Consider a hypothetical hydrogen-like ion where the electron is in the $n=1$ state. If the energy required to ionize this ion from its ground state is four times the ionization energy of a hydrogen atom from its ground state, what is the effective nuclear charge ($Z_{\text{eff}}$) experienced by the electron in this hypothetical ion? Assume the Bohr model is applicable and $S=0$ for the ground state.

    • $1$
    • $2$
    • $4$
    • $\frac{1}{2}$

    Answer: $2$

  4. An element has the electron configuration $[Ar] 3d^{10} 4s^2 4p^3$. What is the effective nuclear charge ($Z_{\text{eff}}$) experienced by a $4p$ electron in this atom, assuming core electrons perfectly screen and valence electrons do not screen each other?

    • $3$
    • $5$
    • $15$
    • $33$

    Answer: $3$

  5. A photon with wavelength $\lambda_1$ has energy $E_1$. If another photon has a wavelength $\lambda_2 = \frac{\lambda_1}{3}$, what is its energy $E_2$ in terms of $E_1$?

    • $E_2 = E_1/3$
    • $E_2 = E_1$
    • $E_2 = 3E_1$
    • $E_2 = 9E_1$

    Answer: $E_2 = 3E_1$

  6. A proton (mass $m_p = 1.672 \times 10^{-27} \text{ kg}$) and an electron (mass $m_e = 9.109 \times 10^{-31} \text{ kg}$) are both accelerated to the same kinetic energy. What is the ratio of the de Broglie wavelength of the proton to that of the electron, $\frac{\lambda_p}{\lambda_e}$?

    • $0.0233$
    • $42.9$
    • $1835$
    • $5.45 \times 10^{-4}$

    Answer: $0.0233$

  7. What is the minimum frequency of light required to ionize a hydrogen atom whose electron is in the $n=3$ excited state? Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$.

    • $3.66 \times 10^{14} \text{ s}^{-1}$
    • $2.42 \times 10^{14} \text{ s}^{-1}$
    • $1.09 \times 10^{15} \text{ s}^{-1}$
    • $3.29 \times 10^{15} \text{ s}^{-1}$

    Answer: $3.66 \times 10^{14} \text{ s}^{-1}$

  8. A baseball ($m = 0.145 \text{ kg}$) is thrown with a velocity of $30 \text{ m/s}$. If the uncertainty in its velocity is $0.10 \text{ m/s}$, what is the minimum uncertainty in its position? Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$.

    • $3.63 \times 10^{-33} \text{ m}$
    • $1.52 \times 10^{-34} \text{ m}$
    • $6.63 \times 10^{-34} \text{ m}$
    • $1.82 \times 10^{-32} \text{ m}$

    Answer: $3.63 \times 10^{-33} \text{ m}$

  9. Which of the following sets of quantum numbers ($n, l, m_l, m_s$) is valid for an electron in a $4f$ subshell, and what is the maximum number of electrons that can occupy all $4f$ orbitals?

    • $(4, 3, -4, +1/2)$; Max electrons: $10$
    • $(4, 3, 0, -1/2)$; Max electrons: $14$
    • $(4, 2, 1, +1/2)$; Max electrons: $14$
    • $(3, 3, -2, -1/2)$; Max electrons: $10$

    Answer: $(4, 3, 0, -1/2)$; Max electrons: $14$

  10. An electron in a hydrogen atom absorbs a photon and transitions from the $n=1$ state to an excited state. If the absorbed photon has a wavelength of $97.2 \text{ nm}$, what is the principal quantum number ($n_f$) of the excited state? Use $h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$ and $c = 3.00 \times 10^8 \text{ m/s}$.

    • $n_f = 2$
    • $n_f = 3$
    • $n_f = 4$
    • $n_f = 5$

    Answer: $n_f = 4$

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