One of the most classic scientific experiments is observing a prism split white light into a rainbow. The phenomenon, known as chromatic dispersion, is the physical principle behind many spectrometers. While prisms were a staple in early spectrometers, they have largely been replaced by diffraction gratings due to their limitations. A prism's ability to disperse incident light is a complex function of three key factors:

• i, the angle of incidence of the light beam on the prism face. This angle, along with the prism's geometry, dictates the path of the refracted light.
• $n$, the refractive index of the prism's material, which is not a constant but varies with wavelength. This property is described by equations like the Cauchy or Sellmeier equations and is the fundamental cause of dispersion.
• $A$, the apex angle of the prism, which determines the overall path length and separation of the dispersed light.

A spectrometer that uses a prism operates on the following principle: First, a lens is used to collimate the light, ensuring the light rays are parallel. This collimated light then passes through a narrow slit to create a well-defined, thin beam. This beam then strikes the prism face at a specific angle of incidence, $i$.

Due to the chromatic dispersion property of the prism material, the refractive index ($n$) is different for each wavelength of light. This causes each constituent color to refract at a slightly different angle, effectively splitting the white light into its individual spectral constituents. These separated wavelengths can then be viewed on a screen or captured by a detector array.

A key limitation of glass prisms is that the variation of their refractive index with wavelength is relatively small and non-linear. This results in poor dispersion and, consequently, low spectral resolution, meaning the device struggles to differentiate between closely spaced wavelengths. The optical resolution—a measure of the instrument's ability to resolve two adjacent spectral lines—dictates the required pixel size or interval for a line array detector. This, in turn, establishes the system's digital resolution.

As you can see in this demonstration of spectral dispersion using a white light LED and a prism, the resulting spectrum is not very sharp and the resolution is low. 

The non-linear dispersion also means the spectrum is compressed at one end and stretched at the other, making uniform calibration difficult. This is precisely why commercial spectrometers rarely, if ever, employ a prism in their design today.