05 Quantum Mechanics and Atomic Orbitals Online Quiz Questions

A particle's momentum changes from $p$ to $2p$, while Planck's constant remains unchanged. Using the de Broglie relation, determine how its wavelength changes and explain why.

An electron drops from a higher allowed energy state to a lower one, and the positive energy gap is $\Delta E$. Give the emitted photon's wavelength in terms of $h$, $c$, and $\Delta E$, and state whether the electron absorbs or emits the photon.

A diagram shows an orbital as a closed surface around a nucleus. Explain what the orbital represents, what $|\psi|^2$ tells you, and why the surface should not be interpreted as a solid boundary that confines the electron absolutely.

A proposed electron state is $(n,l,m_l,m_s)=(3,2,-3,+\frac{1}{2})$. Determine whether it is allowed. Identify the violated rule, if any, and give one valid state obtained by changing only one value.

For a subshell with $l=3$, list the allowed $m_l$ values, determine how many orbitals it contains, and calculate its maximum number of electrons. Explain how the quantum-number rules support both counts.

Determine the numbers of angular, radial, and total nodes in a $4d$ orbital. Show how each count follows from its quantum numbers.

A p subshell has $l=1$. Determine its possible $m_l$ values and explain what these values imply about the number and spatial relationship of its orbitals. Do not assign a specific value to a named Cartesian axis.

Using $E_n=-2.18\times10^{-18}\,\text{J}\,\frac{Z^2}{n^2}$, compare the energy of the $n=2$ state in a one-electron species with $Z=2$ to that of hydrogen at $n=1$. Calculate both energies and explain why they do or do not match.

A student claims that $2s$ and $2p$ orbitals must have the same energy in every atom because they share $n=2$. Evaluate the claim by comparing hydrogen with a multi-electron atom and explaining the role of electron–electron interactions.

A lab report lists an electron using four quantum numbers. Explain what each of $n$, $l$, $m_l$, and $m_s$ describes, and identify which numbers specify the orbital rather than the electron's spin state.

Two electrons occupy the same orbital. Identify which quantum numbers must match between them and which quantum number must differ. State the allowed values that distinguish their spin states and explain how this accounts for the orbital's electron capacity.

An atom has three allowed electron energies, $E_A < E_B < E_C$. Compare the C-to-A and C-to-B transitions: state whether each emits or absorbs a photon, compare the energies and wavelengths of the photons, and explain why. Also state whether the reverse A-to-C transition emits or absorbs a photon.

A particle's momentum increases from p to 4p. Using the de Broglie relation, determine how its wavelength changes and explain the relationship that leads to your result.

At one location in an atom, the wavefunction ψ is zero; at another, |ψ|² is nonzero. Explain what each fact says about the probability density of finding the electron, and explain why neither fact makes an orbital a fixed path.

A proposed electron state has quantum numbers n = 3, l = 2, m_l = +3, and m_s = +1/2. Determine whether the tuple is valid. Identify any invalid value, state its allowed values for the other listed quantum numbers, and give one valid repair that changes as little as possible.

For an electron with n = 4 and l = 1, identify the subshell, list every allowed m_l value, determine the number of orbitals in that subshell, and find its maximum electron capacity. Explain how the quantum-number rules support your results.

For a 4d orbital, calculate the numbers of angular, radial, and total nodes. Show how the counts are related and identify which quantum number determines the angular-node count.

A diagram shows an orbital as a closed surface. A student claims that the electron cannot be found outside it because the surface is the orbital's physical boundary. Correct the claim by explaining what the surface represents and what can be said about the probability outside it.

For a hydrogen atom or another one-electron species with fixed nuclear charge number Z, compare the energies of the n = 2 and n = 4 states. Use the supplied energy dependence on Z and n to express E₄ in terms of E₂, identify which energy is higher, and calculate the energy increase from n = 2 to n = 4 in terms of Z.

A student concludes that 2s and 2p must have equal energies in every atom because they share n = 2. Assess this conclusion separately for hydrogen and for a multi-electron atom, explaining the relevant energy dependence and why the cases differ.

Determine the maximum number of electrons in the n = 3 shell using the quantum-number rules. Show the allowed l values, the number of orbitals for each corresponding subshell, and how the orbital count leads to the shell capacity.

Compare a one-electron species with Z = 1 in the n = 2 state with a one-electron species with Z = 2 in the n = 4 state. Use the energy formula to determine whether the states have equal energies, and calculate their shared energy in joules. Explain the comparison, including the sign.

A student proposes that an electron can settle at an energy halfway between two allowed atomic energy states after absorbing part of a photon. Explain why this proposal conflicts with quantization, and describe the energy relationship required for an absorption or emission transition.

A 5d orbital has $n=5$ and $l=2$. Determine its numbers of angular, radial, and total nodes, and show how you obtain each value.

For a 4d subshell, list the allowed values of $m_l$, determine how many orbitals those values represent, and find the maximum number of electrons the subshell can hold. Explain your reasoning.

Three proposed electron states are listed by $(n,l,m_l,m_s)$: A: $(3,2,-2,+\tfrac12)$; B: $(2,2,0,-\tfrac12)$; C: $(4,1,+2,+\tfrac12)$. Identify which state is allowed. For each disallowed state, identify the quantum number that violates its allowed range and explain why.

Two atomic transitions emit photons with wavelengths of 400 nm and 600 nm. Which transition has the larger electron energy drop, and what is the ratio of the larger photon energy to the smaller photon energy? Explain using $E=hc/\lambda$.

Using $E_n=-2.18\times10^{-18}\,\text{J}\,\frac{Z^2}{n^2}$, compare the orbital energy of $\mathrm{He}^+$ at $n=2$ with that of hydrogen at $n=1$. State each energy in joules and determine whether they are equal.

A student argues that 2s and 2p orbitals must have equal energies in every atom because they share the same principal quantum number. Evaluate the argument for hydrogen and for a multi-electron atom, and explain why the energy pattern can differ.

Imagine electrons are detected one at a time as localized spots, but after many detections the spots form an interference pattern. Explain what each observation indicates about electron behavior and why the combined evidence is not adequately described by treating electrons as only classical particles or only classical waves.