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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?

Maxwell Speed Distribution Curve

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    $\begingroup$ 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. $\endgroup$
    – Poutnik
    Apr 24 '21 at 5:19
  • $\begingroup$ Please elaborate. $\endgroup$ Apr 24 '21 at 6:17
  • $\begingroup$ What part do you not understand ? $\endgroup$
    – Poutnik
    Apr 24 '21 at 6:22
  • $\begingroup$ 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? $\endgroup$ Apr 24 '21 at 6:28
  • $\begingroup$ 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. $\endgroup$
    – Poutnik
    Apr 24 '21 at 8:00

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