How do I use the fact that each element has a different set of energy levels to explain why the colour of the light emitted by an element in the gaseous state is characteristic of the element?

Energy affects frequency and frequency determines wavelenght. The color of an element depends on the wavelength.

Equations for further Proof:
E = hv
(Energy) = (Planck's constant)(frequency)

[[lambda]] = h/mv
wavelength = (Planck's constant)/(mass)(frequency)

To understand why the color of light emitted by an element in the gaseous state is characteristic of that element, we need to consider the concept of energy levels.

In an atom, electrons exist in specific energy levels around the nucleus. These energy levels are quantized, meaning that electrons can only occupy certain discrete energy states. The energy of an electron increases as it moves further away from the nucleus.

When an electron in an atom absorbs energy, it can be excited to a higher energy level. This energy can come from various sources, such as heat or an electrical discharge. However, the excited electron is unstable and tends to return to its original energy level.

When the excited electron returns to a lower energy level, it releases the excess energy in the form of light. The energy of the emitted light is directly related to the energy difference between the higher and lower energy levels involved in this transition. This energy difference determines the frequency and, therefore, the wavelength of the emitted light.

Each element has a specific arrangement of electrons with different energy levels. Since different elements have different numbers of electrons and different electron configurations, their energy level spacings are unique. As a result, the energy differences between the excited and lower energy levels are specific to each element.

The equation E = hv relates the energy (E) of light to its frequency (v), where h represents Planck's constant. Thus, the greater the energy difference between the energy levels, the higher the frequency of the emitted light.

Additionally, the equation λ = h/mv relates the wavelength (λ) of light to the mass (m) and velocity (v) of a particle. While this equation is more applicable to particles with mass, it does not directly determine the color of light emitted by an element.

In summary, the fact that each element has a different set of energy levels in its atoms leads to unique energy differences between these levels. As a result, the emitted light will have characteristic frequencies and wavelengths, which determine its color.