Qualitative Sketch of Radial Probability Distribution for Hydrogen Atom 1s Electron

To find the qualitative plot illustrating the variation of radial probability $P$ with respect to distance $r$ from the nucleus for a 1s electron in a hydrogen atom, we use quantum mechanics principles:

1. Wavefunction of $1\text{s}$ electron: The normalized radial wavefunction $\psi_{1s}(r)$ for the $1\text{s}$ electron of a hydrogen atom is given by: $\psi_{1s}(r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$ where $a_0$ is the Bohr radius.

2. Probability of finding the electron: The probability $P$ of finding the electron within a thin spherical shell of thickness $dr$ at a distance $r$ from the nucleus is the product of the probability density $|\psi_{1s}(r)|^2$ and the volume of the shell $dV = 4\pi r^2 dr$: $P = |\psi_{1s}(r)|^2 \cdot dV = \left(\frac{1}{\pi a_0^3} e^{-2r/a_0}\right) (4\pi r^2 dr)$

$P = \frac{4}{a_0^3} r^2 e^{-2r/a_0} dr$

1. Analyzing the function $P(r)$:

• At $r = 0$: $P(0) = 0$ This indicates that the probability of finding the electron in a shell of radius zero is zero.

• As $r \to \infty$: $\lim_{r \to \infty} P(r) = 0$ The probability decreases asymptotically towards zero as $r$ becomes very large due to the exponential decay factor $e^{-2r/a_0}$.

• Maximum Probability (Most probable radius): To find the value of $r$ where $P$ reaches its maximum, we set the derivative of the radial probability density with respect to $r$ to zero: $\frac{d}{dr} \left( r^2 e^{-2r/a_0} \right) = 0$ $2r e^{-2r/a_0} + r^2 \left(-\frac{2}{a_0}\right) e^{-2r/a_0} = 0$ $2r \left(1 - \frac{r}{a_0}\right) e^{-2r/a_0} = 0$

  Solving for $r \neq 0$: $r = a_0$ Thus, the radial probability $P$ reaches a single maximum peak at $r = a_0$ (the Bohr radius).

1. Conclusion:

• The graph must start at the origin $(0,0)$.
• It rises to a single maximum peak at $r = a_0$.
• It decays exponentially and approaches zero asymptotically as $r \to \infty$ without touching the horizontal axis at any finite distance $r > 0$.

This behavior is correctly represented in plot (D).

Correct Option: D