Where does the maximum electron density occur for 2s and 2p orbitals in hydrogen atom?

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Where does the maximum electron density occur for 2s and 2p orbitals in hydrogen atom?

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Answer 1 · 2015-12-28T09:45:38

For hydrogen, we have to use spherical harmonics, so our dimensions are written as $(r, θ, ϕ)$ $(r, theta, phi)$ . The wave function is defined as follows, via separation of variables:

$ψ_{n l m_{l}} (r, θ, ϕ) = R_{n l} (r) Y_{l}^{m_{l}} (θ, ϕ)$ $color(green)(psi_(nlm_l)(r,theta,phi) = R_(nl)(r) Y_(l)^(m_l)(theta, phi))$

$R_{n l} (r)$ $R_(nl)(r)$ is the radial component of the wave function $ψ_{n l m_{l}} (r, θ, ϕ)$ $psi_(nlm_l)(r,theta,phi)$ , $Y_{l}^{m_{l}} (θ, ϕ)$ $Y_(l)^(m_l)(theta,phi)$ is the angular component, $n$ $n$ is the principal quantum number, $l$ $l$ is the angular momentum quantum number, and $m_{l}$ $m_l$ is the projection of the angular momentum quantum number (i.e. $0, \pm l$ $0, pm l$ ). The wave function represents an orbital.

If you don't understand all of that, that's fine; it was just for context.

To get the maximum electron density, you have to look at probability density curves.

If we plot $4 π r^{2} R_{n l} {(r)}^{2}$ $4pir^2R_(nl)(r)^2$ against $r$ $r$ , we get the probability density curves for an atomic orbital.

The $2 s$ $2s$ orbital's plot looks like this:

![chemwiki.ucdavis.edu)

From this, you can tell that the maximum electron density occurs near $5 a_{0}$ $5a_0$ (with $a_{0} \approx 5.29177 \times 10^{- 11} m$ $a_0 ~~ 5.29177xx10^(-11) "m"$ , the Bohr radius) from the center of the atom, and $4 π r^{2} R_{20} {(r)}^{2}$ $4pir^2 R_(20)(r)^2$ is about $2.45$ $2.45$ or so.

From this similar diagram, we can compare the $2 s$ $2s$ with the $2 p$ $2p$ orbital:

![faculty.uml.edu)

Here, you should see that the $2 p$ $2p$ orbital has a maximum electron density near about $4 a_{0}$ $4a_0$ from the center of the atom, and the value of $4 π r^{2} R_{21} {(r)}^{2}$ $4pir^2 R_(21)(r)^2$ is perhaps around $2.5$ $2.5$ .

This should make more sense once you realize what the probability density plots of the $2 s$ $2s$ and $2 p$ $2p$ orbitals look like:

2s
![http://cronodon.com/](useruploads.socratic.org)

2p
![http://cronodon.com/](useruploads.socratic.org)

"The density of the [dark spots] is proportional to the probability of finding the electron in that region" (McQuarrie, Ch. 6-6).

Basically, start with a radius of 0, and expand your radius of vision outwards from the center of the orbital, and you should be constructing the probability density curves (radial distribution plots).

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Where does the maximum electron density occur for 2s and 2p orbitals in hydrogen atom?

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