Q: Given the transition in a hydrogen atom from $n=3$ to $n=2$, what is the energy change ($\Delta E$) associated with this transition?

$\Delta E = -2.1799 \times 10^{-18} \left( \frac{1}{2^2} - \frac{1}{3^2} \right)$

Question 1

Which of the following statements best explains why the Bohr model fails for multi-electron atoms?

Accepted Answer

The Bohr model cannot fit the positions and momenta of more than one particle simultaneously due to the electron correlation problem.

Question 2

According to the uncertainty principle, which of the following is true about the position and momentum of an electron?

Accepted Answer

There is a fundamental limit to how precisely both position and momentum can be known simultaneously.

Question 3

What is the correct formula for the energy of an electron in a hydrogen-like atom according to the Bohr model?

Accepted Answer

$E_{elec} = -R_H \frac{Z^2}{n^2}$

Question 4

Given the transition in a hydrogen atom from $n=3$ to $n=2$, what is the energy change ($\Delta E$) associated with this transition?

Accepted Answer

$\Delta E = -2.1799 	imes 10^{-18} \left( \frac{1}{2^2} - \frac{1}{3^2} ight)$

Question 5

Which quantum number determines the shape of an atomic orbital?

Accepted Answer

Angular momentum quantum number ($l$)

Question 6

What does the wavefunction $\Psi$ represent in the Schrödinger equation?

Accepted Answer

The amplitude of an electron wave at a given position in space.

Question 7

Which of the following best describes the physical meaning of $|\Psi|^2$ in quantum mechanics?

Accepted Answer

The probability density of finding an electron in a particular region of space.

Question 8

For a hydrogen atom, what is the wavelength of light emitted when an electron transitions from $n=3$ to $n=2$?

Accepted Answer

$\lambda = 656 	ext{ nm}$

Question 9

Which quantum number specifies the orientation of an orbital in space?

Accepted Answer

Magnetic quantum number ($m_l$)

Question 10

Why is the Schrödinger equation considered a more accurate model for atomic structure than the Bohr model?

Accepted Answer

It incorporates the probability of finding electrons and can be extended to multi-electron atoms using approximations.

Question 11

Which of the following is a consequence of the Pauli Exclusion Principle?

Accepted Answer

No two electrons in an atom can have the same set of quantum numbers.

Question 12

What is the effect of electron shielding in multi-electron atoms?

Accepted Answer

It decreases the effective nuclear charge experienced by outer electrons.

Question 13

Which quantum number is associated with the size and energy of an orbital?

Accepted Answer

Principal quantum number ($n$)

Question 14

What is the mathematical form of the time-independent Schrödinger equation for a hydrogen atom?

Accepted Answer

$\hat{H}\Psi = E\Psi$

Question 15

Which of the following best describes the difference between $s$-, $p$-, and $d$-orbitals?

Accepted Answer

$s$-orbitals have $l=0$, $p$-orbitals have $l=1$, and $d$-orbitals have $l=2$.

Question 16

If the energy difference between two levels in a hydrogen atom is $\Delta E = 3.03 	imes 10^{-19} 	ext{ J}$, what is the wavelength of the emitted photon?

Accepted Answer

$\lambda = \frac{hc}{\Delta E}$

Question 17

Which of the following is NOT a quantum number used to describe the state of an electron in an atom?

Accepted Answer

Atomic mass number ($A$)

Question 18

What is the significance of the radial probability density function $D(r)$ in atomic theory?

Accepted Answer

It gives the probability of finding an electron at a specific distance from the nucleus.

Question 19

Which statement about the solutions to the Schrödinger equation for the hydrogen atom is correct?

Accepted Answer

They are quantized and correspond to discrete energy levels.

Atomic Structure: Bohr Model, Quantum Mechanics, and Electron Configuration

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