Question

The lowest energy of an electron confined to a 3-dimensional box of length \(0.5 \, \text{\AA}\) is:

• A. \(E = 2.04 \times 10^{-17} \, \text{J}\)
• B. \(E = 17.24 \times 10^{-17} \, \text{J}\)
• C. \(E = 7.24 \times 10^{-17} \, \text{J}\)
• D. \(E = 1.24 \times 10^{-17} \, \text{J}\)

Hint:

Remember that smaller confinement dimensions lead to higher quantized energy levels due to the L^2 dependence in the denominator.

Answer 1

The Correct Option is C

The energy of an electron confined in a box is given by:
\[E = \frac{h^2}{8mL^2}\]
where $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$, $m = 9.11 \times 10^{-31} \text{ kg}$, and $L = 0.5 \text{ \AA} = 0.5 \times 10^{-10} \text{m}$.
Substituting values:
\[E = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.11 \times 10^{-31} \times (0.5 \times 10^{-10})^2} \approx 7.24 \times 10^{-17} \text{ J}\]