Final answer:
To determine the energy of the electron in a quantum box with one node at the center, we calculate the electron's energy at the second energy level using the quantum particle in a box formula and convert the result to electron volts.
Step-by-step explanation:
The question is asking about the energy of an electron confined in a box with a single node in the center, which is a scenario that relates to the quantum particle in a box model in quantum mechanics, a topic in physics. To find the energy of the electron, we use the formula for the energy levels of a particle in a one-dimensional box:
En = \( \frac{n²h²}{8mL²} \)
where:
n is the principal quantum number,
h is the Planck's constant \( (6.626 \times 10⁽⁻³⁴⁾ J \cdot s) \),
m is the mass of the electron \( (9.109 \times 10⁽⁻³¹⁾ kg) \), and
L is the length of the box
Since there is a single node in the center of the box, the electron is in the second energy level (n=2), as nodes correspond to the n-1 rule. We convert the length of the box to meters (100 pm = 100 x 10-12 m) and plug these values into the formula to calculate the energy. To obtain the energy in electron volts (eV), we convert the result from joules by dividing by the charge of one electron \( (1.602 \times 10⁽⁻¹⁹⁾ C) \).
The energy of the electron in the box is then calculated and presented in eV.