# Radial Probability Distribution Curve versus ψ² versus r curve for 1s orbitals

My question is basic, but I have already referred to a couple of books.

This is an excerpt from Linus Pauling's book:

the most probable distance of the electron from the nucleus, which is the value of $$r$$ at which $$D(r)$$ has its maximum value, is seen from Figure 21-1 to be $$a=\pu{0.529 \mathring{A}}$$, which is just the radius of the normal Bohr orbit for hydrogen.
[Figure 21-1 is the Radial Probability Distribution Curve (RPDC) for the 1s orbital, see below.]

And on the next page he states that:

The function $${\psi^2_{100}}$$ has its maximum value at $$r=0$$, showing that the most probable position for the electron is in the immediate neighborhood of the nucleus; that is, the chance that the electron lie in a small volume element very near the nucleus is larger than the chance that it lie in a volume element of the same size at a greater distance from the nucleus.

Don't these two statements contradict each other? $$\psi^2$$ versus $$r$$ says that the probability density is highest at $$r=0$$. RPDC says that the probability is highest at $$r=\pu{0.529 \mathring{A}}$$. Why is there such a stark difference between probability density and probability? We are varying radius which means that when $$r$$ tends to $$0+$$ we have to have a high electron density which is not the case with respect to the RPDF curve, but it is so with respect to the $$\psi^2$$ versus $$r$$ curve.

I do understand, however, that Radial Probability Distribution Curve represents probability and $$\psi^2$$ versus $$r$$ represents probability density. But why is there a difference in these two?

For reference:

• This is almost certainly fully discussed in (and quite possibly a duplicate of) chemistry.stackexchange.com/questions/104430/…; chemistry.stackexchange.com/questions/92244/…; chemistry.stackexchange.com/questions/107097/… – orthocresol Jul 17 at 16:41
• I do kind of get the language in the answers but the math is kinda advanced for me as a high schooler to understand. Could you please try and word it down for me by reducing the math a bit? That would help a lot. Thanks in advance @orthocresol – Seshank K Jul 17 at 16:51
• And also I agree to all these answers completely with respect to the math. After a while of reading I've almost understood the math entirely but I still have a doubt as to why doesn't chemistry adhere to the mathematics of the equation. Math states that the RPDC curve and psi^2 are same but the same does not hold true when it comes to the arrangement of electrons whereby the electrons are more concentrated closer to r tends to 0+ which is chemically acceptable in the RPDC but mathematically unacceptable. Why? – Seshank K Jul 17 at 16:57
• Correction: "Math states that the RPDC curve and psi^2 are same". What I meant to say was Math says that the graphs are acceptable just the way they are but the chemistry behind the arrangement does not. – Seshank K Jul 17 at 17:20

It is very important to note here that $$R(r)$$ is just the radial part of wavefunction. Wavefunction does not describe any observable. The Born interpretation says that the probability density of finding electron between any two points $$x_1$$ and $$x_2$$ is given by $$\rho=\psi\psi^*.$$ While the probability is $$P =\int_{x_1}^{x_2}{\psi\psi^* \mathrm{d}x}.$$ (This is only in one dimension. In spherical coordinates you must integrate over the volume).
Why is there a difference between these two quantities? I think it will be easier to understand using an analogy. Consider a rod of unit length in which the charge varies as a function of $$x$$. Say the function is $$q(x)=x\mathrm{e}^x-1$$ The average charge of the rod is $$0$$. While the average charge density of this rod is $$\mathrm{e}-2$$. Where average is given by $$\langle f \rangle=\frac{\int^{b}_{a}{f \mathrm{d}x}}{b-a}.$$ I hope you have understood the difference between function density and the function itself. The radial probability and probability density are something similar. By seeing the graphs you can to a certain extent predict the average of the probability density and probability. In the second graph it is $$0$$ while in the second graph it is close to the peak which is $$a_0$$.