# Atomic Spectra and the Quantum Atom Chemistry I · Atoms, Bonds and Reaction Rates · https://tryals.app/learn/chemistry-i/atomic-spectra-and-the-quantum-atom ## Light Comes in Pieces Electromagnetic radiation is characterised by wavelength $\lambda$, frequency $\nu$ and speed $c$, tied together by $c = \lambda\nu$. Classical physics treated radiation as a continuous wave, and then failed twice. The **photoelectric effect** was the first failure. Shining light on a metal ejects electrons, but only above a threshold frequency; below it, no electrons appear no matter how intense the beam. Einstein resolved this by treating light as **photons** of energy $$E = h\nu$$ with $h = 6.626 \times 10^{-34}$ J s. Ejection needs one photon energetic enough to overcome the work function $\phi$; the leftover becomes kinetic energy, $KE = h\nu - \phi$. Intensity sets how *many* photons arrive, not how energetic each one is. The second failure was **atomic spectra**. A heated gas emits only certain sharp wavelengths, and absorbs exactly those same ones. For hydrogen the lines obey the Rydberg formula $$\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$ with $R_H = 1.097 \times 10^7$ m$^{-1}$. **Bohr** explained this by quantising the electron's energy: $$E_n = -\frac{2.18 \times 10^{-18}}{n^2} \text{ J}$$ A photon is emitted only when an electron drops between two allowed levels, so its energy, and therefore its wavelength, can take only particular values. The line spectrum is a direct fingerprint of the ladder of allowed energies. Bohr's model works beautifully for hydrogen and fails for every atom with more than one electron, because it kept classical orbits while bolting quantisation on by hand. > **Common pitfall:** believing brighter light carries more energy *per photon*. Intensity multiplies the number of photons; only frequency changes what a single photon can do. A blinding red lamp will never eject an electron that a faint blue one ejects easily. ## Practice questions 8 of this lesson's 13 practice questions, with answers. The full set is in the app. ### 1. Below a certain frequency, no electrons are ejected from a metal no matter how intense the light. What does this establish? A. Light intensity determines the individual energy carried by each arriving photon B. Electrons require a continuous accumulation of wave energy over time to escape C. Low-frequency radiation is completely reflected rather than absorbed by the metal D. Light delivers energy in discrete photons whose energy depends on frequency **Answer:** D. Light delivers energy in discrete photons whose energy depends on frequency **Why:** A continuous wave would let a dim beam accumulate energy until ejection. A hard frequency threshold means ejection is a **single-photon** event, so energy must arrive in quanta of size $h\nu$. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/below-a-certain-frequency-no-electrons-are-ejected-from-a-metal-no ### 2. Hydrogen emits sharp lines rather than a continuous spectrum. What does this reveal about the electron? A. It possesses the same energy across all sample atoms B. It can only occupy particular allowed energy levels C. It is restricted to orbiting at a single fixed radius D. It loses its energy continuously whilst spiralling inward **Answer:** B. It can only occupy particular allowed energy levels **Why:** Each line is one transition, and its energy is a difference between two states. A discrete set of lines means a discrete set of allowed energies, quantisation, not a continuum of possible orbits. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/hydrogen-emits-sharp-lines-rather-than-a-continuous-spectrum-what ### 3. Which observations are explained by treating light as photons rather than as a continuous wave? A. Instantaneous ejection of electrons at very low intensity B. Emission of sharp spectral lines from a heated gas C. The refraction of light through a glass prism D. A threshold frequency for the photoelectric effect **Answer:** A. Instantaneous ejection of electrons at very low intensity; B. Emission of sharp spectral lines from a heated gas; D. A threshold frequency for the photoelectric effect **Why:** The frequency threshold, sharp line spectra and immediate ejection at low intensity all demand quantised energy packets. Refraction is pure wave behaviour and never needed photons. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/which-observations-are-explained-by-treating-light-as-photons-rather ### 4. An electron falls from $n = 3$ to $n = 2$ in a hydrogen atom. Using $E_n = -2.18 \times 10^{-18}/n^2$ J, compute the emitted photon energy in units of $10^{-19}$ J, to two decimal places. **Answer:** 3.03 (within ±0.06) **Why:** $\Delta E = 2.18 \times 10^{-18}(1/2^2 - 1/3^2) = 2.18 \times 10^{-18} \times 0.1389 = 3.03 \times 10^{-19}$ J, the red H-alpha line at 656 nm. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/an-electron-falls-from-n-3-to-n-2-in-a-hydrogen-atom-using-en ### 5. Match each quantity to what it controls. **Answer:** - Photon frequency → The energy of one photon - Beam intensity → The number of photons per second - Work function → The energy cost of escaping the metal - Principal quantum number → Which rung of the energy ladder **Why:** Frequency fixes energy per photon; intensity fixes photon count; the work function is the metal’s escape cost; $n$ labels the allowed energy level. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/match-each-quantity-to-what-it-controls ### 6. Sort each observation by whether the classical wave picture can account for it. **Answer:** - Explained by waves alone: A beam bending as it passes into glass, Interference fringes from a double slit - Requires photons: A sharp threshold frequency for electron ejection, A glowing gas showing only a few discrete wavelengths, A photocurrent that stops dead below a cut-off colour **Why:** Refraction and interference are pure wave behaviour and never needed quantisation. A frequency threshold, discrete spectral lines and instant ejection at low intensity all require energy to arrive in indivisible packets of $h\nu$. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/sort-each-observation-by-whether-the-classical-wave-picture-can ### 7. Light intensity measures photon count rather than individual photon energy. What does this distinction imply for increasing the brightness of a beam that already ejects photoelectrons? A. Electrons are liberated from deeper atomic energy levels B. Each ejected electron acquires greater kinetic energy C. The required threshold frequency shifts towards red D. More electrons are ejected with unchanged maximum speed **Answer:** D. More electrons are ejected with unchanged maximum speed **Why:** Confusing total beam energy with individual photon energy leads to expecting faster electrons or shifted thresholds. In single-photon interactions, multiplying photon arrivals raises only the rate of collision events, leaving the energy balance per ejected electron untouched. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/light-intensity-measures-photon-count-rather-than-individual-photon ### 8. Arrange these wavelengths of light from lowest photon energy to highest. **Answer:** 1. Radio, 1 m 2. Infrared, 1000 nm 3. Red light, 700 nm 4. Blue light, 450 nm 5. Ultraviolet, 200 nm **Why:** Since $E = hc/\lambda$, energy is inversely proportional to wavelength: the metre-scale radio photon is weakest and the 200 nm ultraviolet photon strongest. Page: https://tryals.app/practice/chemistry-i/atomic-spectra-and-the-quantum-atom/arrange-these-wavelengths-of-light-from-lowest-photon-energy-to