Atoms And Nuclei – NEET UG PhysicsPractice Questions & PYQs

Free atomic hydrogen, particularly when generated in excited electronic states, is chemically reactive and physically unstable due to spontaneous radiative de-excitation. Excited hydrogen atoms quickly transition back to lower energy states by emitting photons within nanoseconds to microseconds. This exceptionally brief duration before a transition or chemical recombination limits the persistence of isolated atomic hydrogen.

Radioactive decay proceeds exponentially according to the law $N(t) = N_0 e^{-\lambda t}$, where $\lambda$ represents the decay constant and $N_0$ is the initial population. The half-life is defined as the specific duration in which the remaining quantity of a radioactive substance drops to exactly half of its initial value. Substituting this condition into the decay equation gives $\frac{N_0}{2} = N_0 e^{-\lambda T_{\frac{1}{2}}}$, which simplifies after canceling $N_0$ and taking the natural logarithm of both sides. Solving the resulting expression isolates the half-life as $T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}$.

The frequency of emitted radiation in hydrogen-like transitions is directly proportional to the energy difference between the energy levels, governed by the relation $\nu \propto \left( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right)$. To maximize this frequency, the inverse square difference must be as large as possible, which occurs when transitioning to the lowest principal quantum number from the closest upper level. Among the choices, the jump from $n = 2$ to $n = 1$ yields the largest energy spacing and thus the highest emission frequency compared to larger orbit drops.

The liquid droplet model treats the atomic nucleus as a charged, cohesive fluid drop held together by surface tension forces analogous to a macroscopic liquid drop. When a heavy nucleus absorbs a neutron, it undergoes collective internal oscillations that distort its spherical shape into an elongated configuration. If the disruptive electrostatic repulsion overcomes the surface tension, the droplet pinches in the middle and violently splits into two smaller fragments, accounting for the mechanics of nuclear fission.

Bohr's atomic model postulates that stable electron orbits are restricted to those whose angular momentum is an integer multiple of $\frac{h}{2\pi}$. By substituting the de Broglie wavelength relation $\lambda = \frac{h}{mv}$ into the angular momentum quantization condition $mvr = \frac{nh}{2\pi}$, the terms rearrange directly into the expression $2\pi r = n \lambda$. This geometric relationship reveals that the total circular circumference of the allowed orbit accommodates an exact integer count of complete de Broglie waves.