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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? When $r$ tends to $0+$ we observe high electron probability density (wrt the $\psi^2$ curve) but minimized electron probability (wrt the RPDF 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:

RPDC curve and psi^2 curve for 1s orbital

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    $\begingroup$ 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/… $\endgroup$
    – orthocresol
    Jul 17, 2019 at 16:41
  • $\begingroup$ 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 $\endgroup$
    – Seshank K
    Jul 17, 2019 at 16:51
  • $\begingroup$ 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? $\endgroup$
    – Seshank K
    Jul 17, 2019 at 16:57
  • $\begingroup$ 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. $\endgroup$
    – Seshank K
    Jul 17, 2019 at 17:20
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    $\begingroup$ Does this answer your question? What is the difference between ψ, |ψ|², radial probability, and radial distribution of electrons? $\endgroup$
    – Nisarg Bhavsar
    Aug 7, 2021 at 18:42

2 Answers 2

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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 $a_0$.

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Probability is basically probability density* volume. Now we're taking some concentric shells around the nucleus and discussing the probability of finding the electron in those region. So when you are too close to the nucleus, the volume of the shell is very small. So even if the probability density is high, the value of probability is low. When r increases, volume of the shells increase, resulting in an increase in probability.

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