Question

Consider a hydrogen atom with \(v_k,\ r_k,\) and \(K_k\) denoting the velocity, orbital radius and kinetic energy of the electron in \(k^{\text{th}}\) orbit, respectively. The electron undergoes a transition from the \(n^{\text{th}}\) orbit, emitting radiation corresponding to the Lyman series. Considering \(h\) to be the Planck's constant and \(\epsilon_0\) the permittivity of free space, the correct statement(s) is/are:

• A. Magnitude of change in kinetic energy can be expressed as \[ \frac{h}{4\pi} \left| \frac{nv_n}{r_n}-\frac{v_1}{r_1} \right| \]
• B. Magnitude of change in de Broglie wavelength can be expressed as \[ \frac{e^2}{4\epsilon_0} \left| \frac1{K_n}-\frac1{K_1} \right| \]
• C. Frequency of radiation emitted can be expressed as \[ \frac{e^2}{8\pi\epsilon_0 h} \left( \frac1{r_1}-\frac1{r_n} \right) \]
• D. Magnitude of change in total energy can be expressed as \[ \frac{h}{2\pi} \left| \frac{v_1}{r_1}-\frac{nv_n}{r_n} \right| \]

Hint:

Important Bohr model results: \[ mvr=\frac{nh}{2\pi} \] \[ K=\frac{e^2}{8\pi\epsilon_0 r} \] \[ E=-K \]

Answer 1

The Correct Option is A

Step 1: Use Bohr model relations.
For hydrogen atom: \[ mvr=\frac{nh}{2\pi} \] Kinetic energy: \[ K=\frac12mv^2 \] Also: \[ K=\frac{e^2}{8\pi\epsilon_0 r} \] Total energy: \[ E=-K \]

Step 2: Check option (A).
Using: \[ m=\frac{nh}{2\pi vr} \] Thus: \[ K=\frac12mv^2 = \frac12\left(\frac{nh}{2\pi vr}\right)v^2 \] \[ K= \frac{nhv}{4\pi r} \] Hence: \[ \Delta K = \frac{h}{4\pi} \left| \frac{nv_n}{r_n}-\frac{v_1}{r_1} \right| \] Therefore option: \[ \boxed{\mathrm{(A)\ is\ correct}} \]

Step 3: Check option (B).
de Broglie wavelength: \[ \lambda=\frac{h}{mv} \] Using: \[ K=\frac12mv^2 \] \[ \lambda=\frac{h}{\sqrt{2mK}} \] Thus wavelength varies as: \[ \frac1{\sqrt K} \] not: \[ \frac1K \] Hence: \[ \boxed{\mathrm{(B)\ is\ incorrect}} \]

Step 4: Check option (C).
Energy emitted: \[ h\nu=E_n-E_1 \] Since: \[ E=-\frac{e^2}{8\pi\epsilon_0 r} \] \[ h\nu = \frac{e^2}{8\pi\epsilon_0} \left( \frac1{r_1}-\frac1{r_n} \right) \] Thus: \[ \nu= \frac{e^2}{8\pi\epsilon_0 h} \left( \frac1{r_1}-\frac1{r_n} \right) \] Hence: \[ \boxed{\mathrm{(C)\ is\ correct}} \]

Step 5: Check option (D).
Since: \[ E=-K \] Magnitude of change in total energy: \[ |\Delta E|=|\Delta K| \] But option (D) gives: \[ \frac{h}{2\pi} \left| \frac{v_1}{r_1}-\frac{nv_n}{r_n} \right| \] which is twice the correct value. Hence: \[ \boxed{\mathrm{(D)\ is\ incorrect}} \]

Step 6: Identify the correct statements.
Therefore: \[ \boxed{\mathrm{(A)\ and\ (C)}} \]