HSC Exam Practice

Sample response. The ground state is the lowest possible energy state of the hydrogen atom, corresponding to the quantum number $n = 1$. Its energy is $E_1 = -13.6$ eV. The negative value indicates the electron is bound to the nucleus; at least 13.6 eV must be supplied to remove the electron completely.

Marking notes. 1 mark for defining ground state as the lowest energy / $n = 1$ state; 1 mark for stating $E_1 = -13.6$ eV (accept $-13.6$ eV or $-2.18 \times 10^{-18}$ J).

Sample response. Postulate 1: Electrons can only orbit the nucleus in certain allowed circular orbits (stationary states) and do not radiate energy while in these orbits. Postulate 2: The angular momentum of the electron is quantised: $L = n\hbar$, where $n = 1, 2, 3, \ldots$ Postulate 3: Electrons emit or absorb a photon only when transitioning between allowed orbits; the photon energy equals the energy difference $E_{photon} = |E_i - E_f|$.

Marking notes. 1 mark per postulate correctly stated. Accept paraphrases. Postulate 1 must include “no radiation in orbit”; Postulate 2 must reference quantised angular momentum or quantised orbits; Postulate 3 must reference photon emission/absorption equalling the energy difference.

Sample response. An electron in a circular orbit is continuously accelerating (centripetal acceleration). Classical electromagnetic theory requires that any accelerating charge must emit electromagnetic radiation, losing energy. As the electron loses energy, it would spiral inward toward the nucleus, reaching it in roughly $10^{-11}$ s — but atoms are observed to be stable. Bohr’s Postulate 1 resolves this by asserting that electrons in allowed orbits do not radiate, despite their acceleration — this is a quantum assumption that has no classical justification.

Marking notes. 1 mark for correctly identifying centripetal acceleration as the cause of radiation in the classical model; 1 mark for explaining energy loss leads to inward spiral (instability); 1 mark for identifying Postulate 1 (no radiation in allowed orbits) as the resolution.

Sample response. An emission spectrum is produced when electrons in excited hydrogen atoms drop to lower energy levels, emitting photons of specific wavelengths as bright coloured lines on a dark background (e.g. Balmer series visible lines when electrons drop to $n = 2$). An absorption spectrum is produced when photons of white light pass through cool hydrogen gas; photons whose energy exactly matches an allowed transition are absorbed, exciting electrons to higher levels, producing dark lines at those wavelengths on a continuous background (e.g. Lyman-$\alpha$ at 121.6 nm absorbed as ground-state electrons jump to $n = 2$). In both cases, only discrete wavelengths corresponding to allowed energy-level differences appear, because electron energies are quantised.

Marking notes. 1 mark for correctly describing emission spectrum (downward electron transition, bright lines); 1 mark for naming an associated series (e.g. Balmer); 1 mark for correctly describing absorption spectrum (upward electron transition, dark lines on continuum); 1 mark for naming an associated series or transition (e.g. Lyman).

Sample response. Bohr’s model is a hybrid because it retains classical mechanics (centripetal force = Coulomb force to derive orbit radii) while grafting on quantum ideas (angular momentum quantised in units of $\hbar$; no radiation in stationary orbits) that have no classical basis. An aspect the model cannot explain from within itself: the quantisation rule $L = n\hbar$ is simply postulated without derivation; the model gives no reason why angular momentum takes discrete values.

Marking notes. 1 mark for identifying a classical element used (centripetal/Coulomb force, or classical orbit picture); 1 mark for identifying a quantum element grafted on (no radiation, or quantised angular momentum); 1 mark for stating a limitation the model cannot explain from within (e.g. why $L$ is quantised, or fine structure, or multi-electron atoms).

Sample response. When cool hydrogen gas is illuminated with white light, photons whose energy exactly matches the difference between an allowed energy level and the ground state (or an excited state populated at low temperature) are absorbed by electrons, exciting them to higher levels. All other photons pass through unaffected. Viewed through a diffraction grating or spectroscope, the transmitted light shows a continuous rainbow spectrum with dark absorption lines at the specific wavelengths that were removed, forming an absorption spectrum. The dark lines appear at specific wavelengths because electron energies are quantised — only photons with exactly the right energy can cause a transition.

Marking notes. 1 mark for describing photon absorption raising electrons to higher levels; 1 mark for noting all other photons pass through; 1 mark for explaining dark lines appear because only specific (quantised) energies are absorbed.

Lyman-$\alpha$: $E_{photon}$. $E = hc/\lambda = (6.626\times10^{-34} \times 3.00\times10^8)/(121.6\times10^{-9}) = 1.634\times10^{-18}$ J $= 10.2$ eV.

Lyman-$\beta$ (full working required): $\lambda$. $1/\lambda = 1.097\times10^7(1/1 - 1/9) = 1.097\times10^7 \times 0.8889 = 9.752\times10^6$ m¹. $\lambda = 1.026\times10^{-7}$ m $= 102.6$ nm. $E_{photon} = hc/\lambda = 1.938\times10^{-18}$ J $= 12.1$ eV.

Series limit ($n_i = \infty$): $\lambda$. $1/\lambda = 1.097\times10^7(1 - 0) = 1.097\times10^7$ m¹. $\lambda = 91.2$ nm. $E_{photon} = 13.6$ eV (ionisation energy).

Marking notes. 1 mark for Lyman-$\beta$ $\lambda$ and energy with correct substitution shown; 1 mark for series-limit $\lambda = 91.2$ nm; 1 mark for all three $E_{photon}$ values correct.

Sample response. From the series limit: $E_{ion} = hc/\lambda_{limit} = (6.626\times10^{-34} \times 3.00\times10^8)/(91.2\times10^{-9}) = 2.18\times10^{-18}$ J [1]. This agrees with the accepted value of $2.18\times10^{-18}$ J to three significant figures [1]. Any small discrepancy is due to rounding in $R_H$ or in the series-limit wavelength; the Rydberg formula is derived from the Bohr model using the same constants, so exact agreement is expected within significant figure limits [1].

Marking notes. 1 mark for correctly calculating $E_{ion}$ from the series-limit $\lambda$; 1 mark for comparison with accepted value; 1 mark for accounting for any discrepancy (rounding / sig figs / model assumptions).

Sample response. All Lyman transitions end at $n_f = 1$, the ground state. The energy gap from any higher level to $n = 1$ is at least 10.2 eV (for $n_i = 2$) [1]. This corresponds to a photon wavelength of at most 122 nm, which is well below the lower limit of visible light ($\approx 400$ nm), placing all Lyman lines firmly in the ultraviolet [1].

Marking notes. 1 mark for linking large energy gap (transitions to $n=1$) to high photon energy/short wavelength; 1 mark for explicitly stating this is below the visible threshold (<400 nm) / in the UV.

Sample response. One assumption: the formula applies to a single electron bound to a nucleus of charge $+Ze$ with no other electrons present (single-electron system). Accept also: no relativistic corrections; point nucleus. A type of atom for which it would be inaccurate: any multi-electron atom such as helium or lithium, because electron–electron repulsion shifts the energy levels significantly from the simple $-13.6Z^2/n^2$ eV prediction.

Marking notes. 1 mark for a valid assumption (single electron / no electron–electron repulsion / non-relativistic); 1 mark for naming a specific type of atom where it fails (any atom with two or more electrons, e.g. helium).

Sample Band 6 response. Bohr’s model (1913) provides a partially satisfactory explanation of hydrogen’s atomic structure but is fundamentally incomplete as a general theory of the atom. The evidence supporting the model is compelling for hydrogen specifically. Bohr’s model correctly predicts the wavelengths of all hydrogen spectral lines, including the Balmer series (visible, $n_f = 2$), Lyman series (UV, $n_f = 1$), and Paschen series (IR, $n_f = 3$), to four significant figures. For example, the H$\alpha$ line is predicted at 656.3 nm and measured at 656.3 nm — exact agreement within measurement precision. The model also correctly gives the ionisation energy of hydrogen as 13.6 eV, matching the experimental value. These successes are direct consequences of Bohr’s two key quantum postulates: quantised angular momentum ($L = n\hbar$) gives the correct energy levels $E_n = -13.6/n^2$ eV, and the photon-emission postulate ($E_{photon} = |E_i - E_f|$) gives the correct wavelengths. However, Bohr’s model has significant limitations that prevent it from being a satisfactory general theory. First, it fails entirely for multi-electron atoms: helium (two electrons) has energy levels that deviate substantially from Bohr’s predictions because the model ignores electron–electron repulsion, an interaction that fundamentally modifies orbital energies. Second, high-resolution spectroscopy reveals that each spectral line of hydrogen is a closely spaced doublet (fine structure), arising from electron spin and relativistic corrections — effects Bohr’s circular-orbit model cannot incorporate. Third, the model postulates angular momentum quantisation without deriving it from first principles, making it an unexplained assertion rather than a consequence of a deeper theory. Quantum mechanics, specifically Schrödinger’s wave equation (1926), addresses all three limitations. The wave function replaces the concept of a defined circular orbit with a probability distribution (“orbital”); quantisation emerges naturally from boundary conditions on the standing wave, not as an arbitrary postulate. For multi-electron atoms, the Schrödinger equation can be applied with perturbation theory to include electron–electron repulsion. The inclusion of spin quantum numbers explains fine structure. In summary, Bohr’s model is satisfactory for hydrogen (and hydrogen-like ions) as a quantitative predictive tool, but is not satisfactory as a general theory of atomic structure. Its historical significance lies in being the first model to introduce quantum ideas into atomic physics, establishing the concept of quantised energy levels that all subsequent quantum theories retained.

Marking criteria (7 marks). 1 = identifies and quantifies at least two specific successes of Bohr’s model for hydrogen (correct wavelength predictions, ionisation energy). 1 = explicitly links successes to the physical postulates responsible (quantised orbits / photon emission on transition). 1 = identifies failure 1: multi-electron atoms, with reason (electron–electron repulsion absent from model). 1 = identifies failure 2: fine structure or another valid limitation, with reason. 1 = states that quantum mechanics (Schrödinger) resolves at least one limitation, with a specific mechanism stated (wave functions / quantisation from boundary conditions / spin). 1 = reaches an explicit evaluative judgement (“satisfactory for hydrogen only” or equivalent — not a blanket pass or fail). 1 = response is coherent, uses precise scientific terminology throughout (energy levels, quantisation, photon, transition, wave function, etc.) and integrates evidence with argument.