# Why is the variable on the y-axis of Maxwell's Speed Distribution Curve 'fraction of molecules'?

I came across the following graph in my textbook showing the kinetic energy distribution of the gas molecules in a gas sample. Now, the thing is the textbook is telling that the area under the curve gives us the fraction of molecules with energy E. Suppose $$dN_{E}$$ represents the number of molecules with energy E, $$N_{T}$$ represents the total number of molecules and $$E$$ represents the kinetic energy of the molecules. Then, the fraction of molecules with energy $$E$$ is given by $$\frac{dN_{E}}{N_{T}}$$, right? So this actually represents the variable in the $$y$$-axis. Now, let us take the area of a small strip under the curve. It is given by $$\frac{dN_{E}}{N_{T}} × dE$$ which is clearly not equal to the fraction of molecules with energy $$E$$. Then why is the book telling that the area under the curve gives us the fraction of molecules with energy $$E$$? It clearly does not. I just proved it. If the area under the curve really has to represent the fraction of molecules with energy $$E$$ then instead of fraction of molecules i.e. $$\frac{dN_{E}}{N_{T}}$$ on the $$y$$-axis, the variable in the $$y$$-axis should actually be $$\frac{1}{dE}\frac{dN_{E}}{N_{T}}$$. Now if we calculate the area of a small strip under the curve :

$$\frac{1}{dE}\frac{dN_{E}}{N_{T}} × dE = \frac{dN_{E}}{N_{T}}$$

The quantity $$\frac{dN_{E}}{N_{T}}$$ actually gives the fraction of molecules with energy $$E$$. Thus, my book is wrong, isn't it? Please can someone clarify if what I have done is correct or not. Can someone explain?

• y represents the fraction density. y . dE represents the fraction dN/N. Integral E0 to infinity of y.dE then represents fraction E>E0. y cannot be 1/dE . dN/N, as the bottom dE does not follow the upper dE. Apr 24 '21 at 5:19
• Please elaborate. Apr 24 '21 at 6:17
• What part do you not understand ? Apr 24 '21 at 6:22
• Is the variable in the y-axis $\frac{1}{dE}\frac{dN_{E}}{N_{T}}$ or is it $\frac{dN_{E}}{N_{T}}$ or is it something else? Apr 24 '21 at 6:28
• The function on the graph is not the fraction, but the fraction density. The fraction is get by multiplying by dE, or integrating over E interval. So the former, but rearranged, as it is formally wrong. It is derivative of the cumulative function f(E)=N(E_mol<=E)/N_T. . Your original expression (1/dE).(dN/N).dE creates false impression both dE are the same, mutually clearable differentials. Apr 24 '21 at 8:00