The Four Laws of Thermodynamics

The Boltzmann distribution is the population in each individual energy state. But there may be many different states with the same energy. So to get the population at a particular energy, you actually need to calculate(number of states with energy = E) times (population of a single state with energy = E).The graph that you posted is actually this quantity.- Actually, energy is proportional to the square of the velocity.

---

If I had a million dollars, I'd buy you a monkey.
Haven't you always wanted a monkey?
-- The Barenaked Ladies

Here's a very simplified example.Suppose that at one energy, E1, the Boltzmann distribution predicts that a state will have 100 molecules in it.In a state of greater energy, E2, the Boltzmann distribution predicts that a state will have 50 molecules in it.However, suppose only one state has energy E1, but there are 4 states with energy E2. Then there will be 100 molecules with energy E1, but there will be 4*50 = 200 molecules with energy E2.So, in this case, twice as many molecules will have the greater amount of energy. That is (if the energy was determined by the kinetic energy only), twice as many molecules will have the higher velocity.Hope this helps -- let's see what further questions that you have.

---

If I had a million dollars, I'd buy you a monkey.
Haven't you always wanted a monkey?
-- The Barenaked Ladies

That is one example. The canonical example, usually worked out in undergraduate thermo courses, are molecules in a cubical box. The quantum mechanical development of this situation also predicts a degeneracy of energy states that increases as energy of the state increases, and this gives exactly the Maxwell distribution.

---

If I had a million dollars, I'd buy you a monkey.
Haven't you always wanted a monkey?
-- The Barenaked Ladies