Question 1

What is the wavelength of the K_α line for Mercury? A) 1.94 × 10^-11m B) 5.38 × 10^-12m C) 1.85 × 10^-11m D) 3.02 × 10^-12 m

Accepted Answer

The K_α line for Mercury is emitted when an electron transitions from the L shell to the K shell. The wavelength of this line can be calculated using the Rydberg formula:$$\frac{1}{\lambda} = R_∞( \frac{1}{n_f^2} - \frac{1}{n_i^2})$$where R_∞ is the Rydberg constant, n_f is the final energy level (1 for the K shell), and n_i is the initial energy level (2 for the L shell). Plugging in these values, we get:$$\frac{1}{\lambda} = 1.097 × 10^7 m^{-1}( \frac{1}{1^2} - \frac{1}{2^2}) = 1.097 × 10^7 m^{-1}(1 - \frac{1}{4}) = 8.228 × 10^6 m^{-1}$$Taking the reciprocal of this value gives us the wavelength:$$\lambda = \frac{1}{8.228 × 10^6 m^{-1}} = 1.215 × 10^{-7} m$$This is the wavelength of the K_α line for Mercury.

Question 2

Doubly-ionized lithium is a hydrogen-like ion,in which the nucleus with charge 3e is orbited by a single electron.What wavelength light would be emitted from Li⁺⁺ as the electron dropped from the n = 4 to the n = 3 orbit? A) 16.9 µm B) 1.88 µm C) 470 nm D) 208 nm

Accepted Answer

The energy difference between n=4 and n=3 orbits is given by the formula:
ΔE = 13.6 * Z^2 * (1/nf^2 - 1/ni^2)
where Z = 3 (charge of the nucleus), nf = 3 (final orbit), and ni = 4 (initial orbit).
Plugging in these values, we get:
ΔE = 13.6 * 3^2 * (1/3^2 - 1/4^2) = 5.925 eV
We can use the formula ΔE = hc/λ to find the wavelength of the emitted light, where h is Planck's constant, c is the speed of light, and λ is the wavelength.
Rearranging the formula, we get:
λ = hc/ΔE
Substituting the values, we get:
λ = (6.626 x 10^-34 J.s) * (2.998 x 10^8 m/s) / (5.925 eV * 1.602 x 10^-19 J/eV) = 208 nm
Therefore, the answer is D) 208 nm.

Question 3

An atom in a state with its orbital quantum number l = 1 decays to its ground state (with l = 0).A photon of wavelength 630.000 nm is emitted in the process.When the same process takes place in the presence of an intense magnetic field,the following change in the spectrum is observed.With the magnetic field present,one of the emitted lines observed now has a wavelength of 630.030 nm.Which of the following wavelengths would you expect to be also present?
A) 629.970 nm
B) 630.060 nm
C) 630.090 nm
D) 630.120 nm
E) 629.910 nm

Accepted Answer

The presence of an intense magnetic field causes the Zeeman effect, which splits spectral lines. The original line at 630.000 nm splits into multiple components. Given that one of the split lines is at 630.030 nm, which is 0.030 nm longer than the original, there should be a symmetrically split line 0.030 nm shorter than the original, at 629.970 nm. This symmetry is due to the equal energy difference caused by the magnetic field for the split components.

Question 4

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state? A) 2E₀, single degeneracy B) 2E₀, triple degeneracy C) 3E₀, single degeneracy D) 6E₀, single degeneracy E) 6E₀, triple degeneracy

Accepted Answer

Since the potential is infinite outside of the given region, the wavefunction of the particle must be zero at the boundaries of the box. This implies that the wavefunction must have nodes at the boundaries. The ground state wavefunction will have no nodes (it is just a constant) and thus has zero energy. The first excited state must have one node along each direction, and thus three nodes in total. This gives rise to three possible wavefunctions: one with a node along x, one with a node along y, and one with a node along z. These three wavefunctions have the same energy, and thus there is a triple degeneracy. The energy of the first excited state is therefore 2E₀ (twice the ground state energy).

Question 5

Model a hydrogen atom as a three-dimensional potential well with U₀= 0 in the region ,and 0 < z < L,and infinite otherwise,with L = 1.0 × 10^-10 m.Which of the following is NOT one of the lowest three energy levels of an electron in this model? A) 113 eV B) 226 eV C) 283 eV D) 339 eV

Accepted Answer

The energy levels of an electron in a hydrogen atom are given by the equation: 
En = -13.6 eV/n^2 
where n is the principal quantum number. 
In the potential well model given in the question, the electron is confined to a three-dimensional box with a length of L in the z direction and infinite in all other directions. The energy levels of such a box are given by: 
Enx,y,z = (h^2/8mL^2) (nx^2 + ny^2 + (nz/L)^2) 
where nx, ny, and nz are integers representing the number of nodes in the x, y, and z directions, respectively. 
The lowest three energy levels of this model are: 
E1 = h^2/8mL^2 = 10.2 eV 
E2 = 3h^2/8mL^2 = 40.7 eV 
E3 = 5h^2/8mL^2 = 91.4 eV 
Therefore, the answer is C, since 283 eV is not one of the lowest three energy levels of an electron in this model.

Question 6

The energy of an electron in the p-level of an atom is changed in the presence of a magnetic field of magnitude 4.6 T.What is the difference between the largest and smallest possible energies? (Bohr magneton = μ_B = 9.27 × 10^-24 J/T)

Accepted Answer

The answer of The energy of an electron in the...

Question 7

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state? A) 2E₀, single degeneracy B) (4/3)E₀, single degeneracy C) 2E₀, double degeneracy D) (7/3)E₀, double degeneracy E) 3E₀, triple degeneracy

Accepted Answer

In a three-dimensional potential well with a bound particle, the ground-state energy is given by: E1= 3h^2/(8mL^2), where L is the length of the cube. The first excited state will occur when the particle is excited to the first excited state in one of the dimensions, say the x-dimension. Therefore, the wavefunction will have two nodes along the x-dimension, and one node along each of the y and z-dimensions. The energy of the first excited state can be calculated using the formula: E2= n^2*h^2/(8mL^2), where n=2 (because there is one node along the x-dimension) and L is the length of the cube. Therefore, E2= (4/3)E1. Since there is only one possible way to excite the particle to the first excited state, the degeneracy of this state is single. Hence, the correct choice is (B) (4/3)E₀, single degeneracy.

Question 8

An alkali metal atom is in the ground state.The orbital angular momentum equals zero and the spin angular momentum is entirely due to the single valence electron.A magnetic field is applied that splits the ground state energy level into two levels,65 μeV apart.A photon,absorbed by the atom,induces a transition between the two levels.What is the wavelength of the photon? (c = 3.00 × 10⁸ m/s,h = 6.626 × 10^-34 J • s,Bohr magneton = μ_B = , A) 19 mm B) 25 mm C) 31 mm D) 38 mm E) 41 mm

Accepted Answer

The answer of An alkali metal atom is in the...

Question 9

What is the correct electronic configuration for the ground state sodium atom,which has 11 electrons? A) 1s¹2s²3p⁶2s² B) 1s²2s¹3p⁶2s² C) 1s¹2s²2p⁶ D) 1s²2s²2p⁶3s² E) 1s²2s²2p⁶3s¹

Accepted Answer

The correct electronic configuration for a ground state sodium atom is 1s² 2s² 2p⁶ 3s¹, which corresponds to choice E. Sodium has 11 electrons, and this configuration fills the 1s, 2s, and 2p orbitals completely, and places the last electron in the 3s orbital.

Question 10

A muonic hydrogen atom is a proton orbited by a muon (a particle with the same charge as an electron and 207 times its mass)in which the mass of the muon is significant relative to the mass of the proton.What would be the radius of the smallest muon orbit in a muonic hydrogen atom? A) 1.03 × 10^-8m B) 1.10 × 10^-8m C) 2.56 × 10^-13m D) 2.84 × 10^-13m E) 5.29 × 10^-11m

Accepted Answer

The radius of the smallest electron orbit in a hydrogen atom can be calculated using the Bohr radius formula:
r = (4πε0ħ²)/(me²)
where ε0 is the permittivity of free space, ħ is the reduced Planck constant, and me is the mass of the electron.
For a muonic hydrogen atom, we need to replace me with the reduced mass:
μ = (mp*mμ)/(mp+ mμ)
where mp is the mass of the proton and mμ is the mass of the muon.
Using the same formula, we get:
r = (4πε0ħ²)/(μe²)
Plugging in the values, we get:
r = (4π*8.85×10⁻¹²*1.05×10⁻³⁴/(2.01*1.67×10⁻²⁷*5.69×10⁻28*1.60×10⁻¹⁹)
r = 2.84 × 10⁻¹³ m, which is closest to choice (D).

Question 11

An s state (l = 0)energy level is split into two levels by an applied magnetic field.A photon of microwave radiation having frequency 60 GHz induces a transition between the two levels.What is the magnitude of the applied magnetic field? (h = 6.626 × 10^-34 J • s, A) 2.1 T B) 0.34 T C) 1.2 T D) 2.8 T E) 13 T

Accepted Answer

The answer of An s state (l = 0)energy level...

Question 12

What is the radius of the smallest electron orbit in a singly-ionized helium atom? A) 1.32 × 10^-11m B) 2.65 × 10^-11m C) 5.29 × 10^-11m D) 1.06 × 10^-10m

Accepted Answer

The answer of What is the radius of the smallest...

Question 13

An alkali metal atom is in the ground state.The orbital angular momentum equals zero and the spin angular momentum is entirely due to the single valence electron.A magnetic field is applied that splits the ground state energy level into two levels,27 μeV apart.What is the strength of the applied magnetic field? (h = 6.626 × 10^-34 J ∙ s,Bohr magneton = μ_B = ,1 eV = 1.60 × 10^-19 J) A) 0.23 T B) 0.18 T C) 0.29 T D) 0.34 T E) 0.40 T

Accepted Answer

The answer of An alkali metal atom is in the...

Quiz 41: Atomic Structure

What is the wavelength of the K_α line for Mercury?

Doubly-ionized lithium is a hydrogen-like ion,in which the nucleus with charge 3e is orbited by a single electron.What wavelength light would be emitted from Li⁺⁺ as the electron dropped from the n = 4 to the n = 3 orbit?

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state?

Model a hydrogen atom as a three-dimensional potential well with U₀= 0 in the region ,and 0 < z < L,and infinite otherwise,with L = 1.0 × 10^-10 m.Which of the following is NOT one of the lowest three energy levels of an electron in this model?

The energy of an electron in the p-level of an atom is changed in the presence of a magnetic field of magnitude 4.6 T.What is the difference between the largest and smallest possible energies? (Bohr magneton = μ_B = 9.27 × 10^-24 J/T)

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state?

What is the correct electronic configuration for the ground state sodium atom,which has 11 electrons?

A muonic hydrogen atom is a proton orbited by a muon (a particle with the same charge as an electron and 207 times its mass) in which the mass of the muon is significant relative to the mass of the proton.What would be the radius of the smallest muon orbit in a muonic hydrogen atom?

An s state (l = 0) energy level is split into two levels by an applied magnetic field.A photon of microwave radiation having frequency 60 GHz induces a transition between the two levels.What is the magnitude of the applied magnetic field? (h = 6.626 × 10^-34 J • s,

What is the radius of the smallest electron orbit in a singly-ionized helium atom?

Quiz 41: Atomic Structure

What is the wavelength of the Kα line for Mercury?

Doubly-ionized lithium is a hydrogen-like ion,in which the nucleus with charge 3e is orbited by a single electron.What wavelength light would be emitted from Li++ as the electron dropped from the n = 4 to the n = 3 orbit?

A three-dimensional potential well has potential U0 = 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E0.What is the energy of the first excited state,and what is the degeneracy of that state?

Model a hydrogen atom as a three-dimensional potential well with U0 = 0 in the region ,and 0 < z < L,and infinite otherwise,with L = 1.0 × 10-10 m.Which of the following is NOT one of the lowest three energy levels of an electron in this model?

The energy of an electron in the p-level of an atom is changed in the presence of a magnetic field of magnitude 4.6 T.What is the difference between the largest and smallest possible energies? (Bohr magneton = μB = 9.27 × 10-24 J/T)

A three-dimensional potential well has potential U0 = 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E0.What is the energy of the first excited state,and what is the degeneracy of that state?

What is the correct electronic configuration for the ground state sodium atom,which has 11 electrons?

A muonic hydrogen atom is a proton orbited by a muon (a particle with the same charge as an electron and 207 times its mass) in which the mass of the muon is significant relative to the mass of the proton.What would be the radius of the smallest muon orbit in a muonic hydrogen atom?

An s state (l = 0) energy level is split into two levels by an applied magnetic field.A photon of microwave radiation having frequency 60 GHz induces a transition between the two levels.What is the magnitude of the applied magnetic field? (h = 6.626 × 10-34 J • s,

What is the radius of the smallest electron orbit in a singly-ionized helium atom?

What is the wavelength of the K_α line for Mercury?

Doubly-ionized lithium is a hydrogen-like ion,in which the nucleus with charge 3e is orbited by a single electron.What wavelength light would be emitted from Li⁺⁺ as the electron dropped from the n = 4 to the n = 3 orbit?

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state?

Model a hydrogen atom as a three-dimensional potential well with U₀= 0 in the region ,and 0 < z < L,and infinite otherwise,with L = 1.0 × 10^-10 m.Which of the following is NOT one of the lowest three energy levels of an electron in this model?

The energy of an electron in the p-level of an atom is changed in the presence of a magnetic field of magnitude 4.6 T.What is the difference between the largest and smallest possible energies? (Bohr magneton = μ_B = 9.27 × 10^-24 J/T)

A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state?

An s state (l = 0) energy level is split into two levels by an applied magnetic field.A photon of microwave radiation having frequency 60 GHz induces a transition between the two levels.What is the magnitude of the applied magnetic field? (h = 6.626 × 10^-34 J • s,