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Answer the following.
Show that radius (r_n) of the n^(th) Bohr orbit varies directly with square of the principal…
(a)[2.5]
Show that radius ($r_n$) of the $n^{\text{th}}$ Bohr orbit varies directly with square of the principal quantum number ($n$) of the orbit.
(b)[2.5]
(1) Where does nuclear fusion reaction take place continuously in the Universe?
(2) What is meant by “Mass defect” of a nucleus?
Answer
Answer (a)
Derivation of $r_n \propto n^2$ for the Bohr orbit.
- 1. The electron (mass $m$, charge $e$) moves in a circular orbit of radius $r$ with speed $v$ around a nucleus of charge $Ze$; the Coulomb force supplies the centripetal force: $\frac{Ze^2}{4\pi\epsilon_0 r^2} = \frac{mv^2}{r}$ (Coulomb's law and Newton's second law).
- 2. Bohr's quantisation condition: $mvr = \frac{nh}{2\pi}$, so $v = \frac{nh}{2\pi mr}$.
- 3. Substituting $v$ in step 1: $\frac{Ze^2}{4\pi\epsilon_0 r} = m\left(\frac{nh}{2\pi mr}\right)^2 = \frac{n^2h^2}{4\pi^2 m r^2}$.
- 4. Solving for $r$: $r_n = \frac{n^2 h^2\epsilon_0}{\pi m Z e^2}$.
- 5. Since $h$, $\epsilon_0$, $m$, $Z$ and $e$ are constants for a given atom, $r_n \propto n^2$ (for hydrogen, $r_n = n^2 \times 0.529\text{ Å}$).
Answer (b)
(1) In the core of the Sun and other stars.
(2) Mass defect is the difference between the sum of the masses of the individual nucleons and the actual mass of the nucleus: $\Delta m = [Zm_p + (A-Z)m_n] - M$.