What is the ratio of the density of states for electrons at 2.7 electron volt and at 0.27 electron volts?
This term, density of states, refers to how many different electron states are able to occupy a given volume. This density of states, sometimes represented as a lowercase 𝑔, is a function of the energy of the electron. This density of states is expressed through geometric factors divided by Planck’s constant cubed all multiplied by the mass of the electron to the three-halves power times the square root of the energy of the electron.
Since we want to solve for a ratio of electrons at two different energy levels, we’ll divide the density of states at the higher energy level by the density of states at the lower level. When we enter these expressions, we see that absolutely everything in them cancels out except for the square root of the energies of our respective levels.
When we enter this expression on our calculator, to two significant figures, we find a result of 3.2. That’s the ratio of the density of states for electrons at 2.7 electron volts and at 0.27 electron volts.