(b) the idea that energy levels have negative values

The idea

Energy levels in an atom are quoted as negative values because the zero of energy is defined as the electron being just free of the nucleus, at rest and infinitely far away. A bound electron is held by the electrostatic attraction, so removing it requires energy input; the bound state therefore sits below the zero line and its energy is negative. For example, the ground state of hydrogen is about −13.6 eV, and an electron excited to a higher level has a less negative value such as −3.4 eV. The energy needed to ionise the atom equals the difference between zero and the level, so a level of −13.6 eV needs 13.6 eV to free the electron. Transitions between levels involve differences, so the negative signs cancel correctly when you subtract.

What earns the marks

• States that the zero of energy is taken as the electron at rest, infinitely far from the nucleus (ionised).
• Explains that a bound electron is attracted to the nucleus, so energy must be supplied to remove it.
• Links the negative value to the bound state lying below the zero reference level.
• Uses a numerical example such as the hydrogen ground state at about −13.6 eV to show a negative level.
• Recognises that the ionisation energy equals the difference between zero and the level, giving a positive value.
• Applies subtraction of two negative level values to find the photon energy for a transition.

Common mistakes and how to put them right

• Thinking a negative energy level means the electron has negative kinetic energy.

  Put it right: The negative sign is a bookkeeping result of choosing the free electron as zero, and kinetic energy of a bound electron is still positive.

• Believing that a more negative level is a higher energy state.

  Put it right: A more negative value is a more tightly bound, lower energy state, so −13.6 eV is below −3.4 eV.

• Adding the magnitudes of two negative levels instead of subtracting them.

  Put it right: The photon energy is the difference, for example −3.4 eV minus −13.6 eV gives 10.2 eV.

• Assuming the zero level is the nucleus or the ground state.

  Put it right: The zero is the electron fully removed and at rest at infinity.

Exam tips

• Sketch a vertical energy-level diagram with zero at the top and negative values below, then mark the transition arrow downwards for emission.
• When calculating a transition energy, write the subtraction in full with both signs before evaluating, so sign errors are visible.
• Check that any ionisation energy you quote is positive, since it is an energy input, even though the level itself is negative.