AP Physics 2 Exam Question: Atomic Structure
Consider an atom of hydrogen that is in its ground state. The electron in this atom is in the n=1 energy level. Assume that the mass of the hydrogen atom is approximately 1 u (atomic mass unit) and that atomic units are used. The following three scenarios involve the electron transitioning from its ground state to a higher energy level:
Scenario 1: The electron absorbs a photon with a wavelength of 121.6 nm.
Scenario 2: The electron absorbs a photon with a wavelength of 244.0 nm.
Scenario 3: The electron absorbs a photon with a wavelength of 108.2 nm.
a) Determine the energy change (ΔE) in atomic units for each of the above scenarios. Show step-by-step calculations and include the conversions if necessary.
b) Calculate the absolute value of the photon's energy (E) in joules for each scenario (recall that 1 eV = 1.602 x 10^-19 Joules).
c) Discuss the significance of the energy changes (ΔE) and the resulting photon absorption in each scenario, with respect to hydrogen's atomic structure.
Answer:
a) To determine the energy change (ΔE) in atomic units, we can make use of the Rydberg formula:
1/λ = R * (1/n_final^2 - 1/n_initial^2)
where λ is the wavelength of the absorbed photon, R is the Rydberg constant (approximately 1.097 x 10^7 m^-1), n_final is the final energy level, and n_initial is the initial energy level.
Scenario 1: Given λ = 121.6 nm = 121.6 x 10^-9 m and n_initial = 1, we can use the Rydberg formula:
1/λ = R * (1/n_final^2 - 1/1^2)
1.0/121.6 x 10^-9 = 1.097 x 10^7 * (1 - 1/n_final^2)
n_final^2 - 1 = 1.097 x 10^7 * 121.6 x 10^-9
n_final^2 = 1 + 1.097 x 10^7 * 121.6 x 10^-9
n_final^2 ≈ 1.000011
Taking the square root, we find n_final ≈ 1 (since the energy level must be an integer).
Therefore, the energy change (ΔE) in atomic units for Scenario 1 is approximately 0 atomic units.
Scenario 2: Given λ = 244.0 nm = 244.0 x 10^-9 m and n_initial = 1, we can use the Rydberg formula:
1/λ = R * (1/n_final^2 - 1/1^2)
1.0/244.0 x 10^-9 = 1.097 x 10^7 * (1 - 1/n_final^2)
n_final^2 - 1 = 1.097 x 10^7 * 244.0 x 10^-9
n_final^2 = 1 + 1.097 x 10^7 * 244.0 x 10^-9
n_final^2 ≈ 1.001942
Taking the square root, we find n_final ≈ 1 (since the energy level must be an integer).
Therefore, the energy change (ΔE) in atomic units for Scenario 2 is approximately 0 atomic units.
Scenario 3: Given λ = 108.2 nm = 108.2 x 10^-9 m and n_initial = 1, we can use the Rydberg formula:
1/λ = R * (1/n_final^2 - 1/1^2)
1.0/108.2 x 10^-9 = 1.097 x 10^7 * (1 - 1/n_final^2)
n_final^2 - 1 = 1.097 x 10^7 * 108.2 x 10^-9
n_final^2 = 1 + 1.097 x 10^7 * 108.2 x 10^-9
n_final^2 ≈ 1.00002
Taking the square root, we find n_final ≈ 1 (since the energy level must be an integer).
Therefore, the energy change (ΔE) in atomic units for Scenario 3 is approximately 0 atomic units.
b) To calculate the absolute value of the photon's energy (E) in joules, we can use the equation:
E = ΔE * (2π^2 * m_e * e^4) / (h^2)
where ΔE is the energy change in atomic units, m_e is the electron's mass (approximately 9.10938356 x 10^-31 kg), e is the elementary charge (approximately 1.602176634 x 10^-19 C), and h is Planck's constant (approximately 6.62607015 x 10^-34 J·s).
Since the energy change (ΔE) for all scenarios is 0 atomic units, the absolute value of the photon's energy (E) is also 0 Joules for each scenario.
c) The energy changes (ΔE) calculated in parts (a) and (b) indicate that the photons absorbed in all three scenarios don't cause the electron to transition to a higher energy level. The energy levels in hydrogen's atomic structure are quantized, meaning that only certain energy states are allowed. In the case of the hydrogen atom in its ground state, the electronic transition to higher energy levels requires more energy. Therefore, these scenarios do not result in a change in the electron's energy level.