Structure of Atom

Radius and Velocity of n-th Bohr Orbit

Conditions: Circular orbit of an electron around a stationary nucleus of atomic number Z

• Equate electrostatic force to centripetal force: \frac{1}{4\pi \epsilon_0} \frac{Ze^2}{$r^2 = \frac{mv^2}{r}$
• Apply Bohr's quantization condition for angular momentum: mvr = \frac{nh}{2\pi}
• Eliminate velocity v from the two equations to solve for radius r_n
• Substitute r_n back into the velocity equation to get v_n

Result: $r_n = \frac{\epsilon_0 n^2 h^2}{\pi m Z e^2}$ and $v_n = \frac{Ze^2}{2 \epsilon_0 n h}$

Total Energy of Electron in n-th Bohr Orbit

Conditions: Hydrogen-like atom (single electron)

• Calculate kinetic energy $K = \frac{1}{2} mv^2 = \frac{1}{4\pi \epsilon_0} \frac{Ze^2}{2r}$
• Calculate potential energy $U = -\frac{1}{4\pi \epsilon_0} \frac{Ze^2}{r}$
• Sum kinetic and potential energy: $E = K + U = -\frac{1}{4\pi \epsilon_0} \frac{Ze^2}{2r}$
• Substitute the expression for r_n into the total energy equation

Result: $E_n = -\frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2} = -13.6 \frac{Z^2}{n^2} \text{ eV}$

Energy Level Diagram of Hydrogen Atom

Draw: Horizontal energy lines plotted against vertical energy scale in eV, showing converging levels towards 0 eV and spectral series transitions.

Interpret: Illustrates discrete energy states, ionization energy at 13.6 eV, and different series (Lyman in UV, Balmer in visible, Paschen/Brackett in IR).

Exam use: Direct questions on minimum/maximum wavelength of spectral series and excitation energy calculations.

Frank-Hertz Experiment Setup

Draw: Collector current versus accelerating potential graph showing periodic drops corresponding to excitation potentials.

Interpret: Proves the existence of discrete atomic energy levels as postulated by Bohr.

Exam use: Questions on mercury vapor excitation potential (4.9 eV) and inelastic collisions.

Stern-Gerlach Experiment

Draw: Splitting of a beam of silver atoms passing through an inhomogeneous magnetic field into two distinct traces.

Interpret: Demonstrates space quantization and spatial orientation of intrinsic electron spin angular momentum ($m_s = \pm 1/2)$.

Exam use: Conceptual questions on space quantization and spin degrees of freedom.

Calculate the wavelength of the first line of the Balmer series in the hydrogen spectrum. (Given $R = 1.097 x 10^7 m^-1)$

• Identify the transitions for the first line of Balmer series: $n_1 = 2$ and $n_2 = 3$.
• Use the Rydberg formula: 1/$\lambda = R (1/n_1^2 - 1/n_2^2)$.
• Substitute values: 1/$\lambda = 1.097 \times 10^7 (1/2^2 - 1/3^2) = 1.097 \times 10^7 (1/4 - 1/9) = 1.097 \times 10^7 (5/36)$.
• Calculate $\lambda = 36 / (5 \times 1.097 \times 10^7) = 6.563 \times 10^-7 \text{ m}$.

Answer: 656.3 nm (or 6563 Å)

Find the possible values of total angular momentum quantum number J for two electrons with orbital quantum numbers $l_1 = 1$ and $l_2 = 1 under L-S coupling$.

• For $l_1 = 1$ and $l_2 = 1$, total orbital angular momentum L can take values from |l_1 - l_2| to l_1 + l_2, i.e., L = 0, 1, 2.
• For two electrons, individual spins are $s_1 = 1/2$ and $s_2 = 1/2$. Total spin S can take values 0 and 1.
• Total angular momentum J for each (L, S) combination ranges from |L - S| to L + S.
• For L = 0, $S = 0 -> J = 0$. For L = 0, $S = 1 -> J = 1$. For L = 1, $S = 0 -> J = 1$. For L = 1, $S = 1 -> J = 0$, 1, 2. For L = 2, $S = 0 -> J = 2$. For L = 2, $S = 1 -> J = 1$, 2, 3.

Answer: J values are 0, 1, 2, and 3 depending on the L and S state combinations.

What is the radius of the third orbit of a hydrogen atom? (Bohr radius $r_1 = 0.53 Å)$

• Use the radius relation for hydrogen-like atoms: $r_n = r_1 \times (n^2 / Z)$.
• For hydrogen, Z = 1, and for the third orbit, n = 3.
• Substitute the values: $r_3 = 0.53 \times (3^2 / 1) = 0.53 \times 9$.
• Multiply: $r_3 = 4.77 Å$.

Answer: 4.77 Å