Question

How does the energy of a principal energy level depend on the value of \(n\) ? Does a higher value of \(n\) mean a higher or lower energy?

Short answer

The energy of an electron in a principal energy level $n$ of a hydrogen-like atom is given by the formula \(E = -\dfrac{13.6 \, eV}{n^2}\). As $n$ increases, the energy E becomes less negative (closer to zero), indicating a higher energy. Therefore, a higher value of $n$ corresponds to a higher energy, while a lower value of $n$ corresponds to a lower energy.

Step-by-step solution

Energy formula for an electron in a hydrogen-like atom

The energy of an electron in a principal energy level n of a hydrogen-like atom can be calculated using the formula: \[ E = -\dfrac{13.6 \, eV}{n^2} \] Where E is the energy of the electron, and n is the principal quantum number (an integer n = 1, 2, 3, ...).

Analyzing the relationship between energy and n

Looking at the formula, we can notice that as n increases, the denominator \(n^2\) also increases, which makes the entire fraction smaller. Since it's a negative fraction, when the fraction becomes smaller, the overall energy E becomes less negative or closer to zero. When n is small (n = 1, for example), the energy is more negative, and as n gets larger, the energy becomes less negative.

Conclusion: Higher or lower energy for larger n?

From the analysis, we can conclude that a higher value of n means a higher energy as the energy E becomes closer to zero (less negative). Conversely, a lower value of n means lower energy (more negative) in the energy level.