Question

What is the distance of a `2s` electron from the nucleus of H atom?

Answer 1

It is `6` times the Bohr radius, but you have to suffer through a gauntlet of integration by parts.

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The `2s` atomic orbital wave function for `"H"` atom is:

`psi_(2s) = 1/(4sqrt(2pi))(1/a_0)^(3//2)(2 - (r)/a_0)e^(-r//2a_0)`

The average value in general is given by

`<< x >> = int_"allspace" xp(x)dx`,

where `p(x)` would be the probability distribution function.

Since `psi^"*"psi` is the probability density, the average distance from the nucleus, or `<< r >>`, the expectation value for the radial position, is found by solving the following integral:

`<< r >> = int_(0)^(2pi) int_(0)^(pi) int_(0)^(oo) r cdot psi_(2s)^"*"psi_(2s) cdot r^2drsin thetad thetad phi`

`= 1/(32pi)(1/a_0)^(3)int_(0)^(2pi) int_(0)^(pi) int_(0)^(oo) [(2 - (r)/a_0)e^(-r//2a_0)]^2 r^3drsin thetad thetad phi`

`= 1/(32pi)(1/a_0)^(3)int_(0)^(2pi) int_(0)^(pi) int_(0)^(oo) (4 - (4r)/a_0 + (r/a_0)^2)r^3e^(-r//a_0) drsin thetad thetad phi`

`= 1/(32pi)(1/a_0)^(3)int_(0)^(2pi) int_(0)^(pi) int_(0)^(oo) (4e^(-r//a_0) - (4r)/a_0e^(-r//a_0) + (r/a_0)^2e^(-r//a_0)) r^3drsin thetad thetad phi`

The integral `int_(0)^(2pi) int_(0)^(pi) sinthetad thetad phi = 4pi`, so...

`= 1/(8)(1/a_0)^(3) int_(0)^(oo) (4e^(-r//a_0) - (4r)/a_0e^(-r//a_0) + (r/a_0)^2e^(-r//a_0)) r^3dr`

`= 1/(8)(1/a_0)^(3) int_(0)^(oo) 4r^3e^(-r//a_0) - (4r^4)/a_0e^(-r//a_0) + (r/a_0)^2r^3e^(-r//a_0)dr`

`= 1/(8)(1/a_0)^(3) int_(0)^(oo) 4r^3e^(-r//a_0)dr - 1/(8)(1/a_0)^(3) int_(0)^(oo)(4r^4)/a_0e^(-r//a_0)dr + 1/(8)(1/a_0)^(3) int_(0)^(oo)(r/a_0)^2r^3e^(-r//a_0)dr`

I'm not going to go past this though, because it is just ugly integration by parts. Using Wolfram Alpha...

• The first integral gives `3a_0`.
• The second gives `12a_0`.
• The third gives `15a_0`.

So, the result is:

`color(blue)(<< r >>) = 3a_0 - 12a_0 + 15a_0 = color(blue)(6a_0)`

And to check, someone on stackexchange got a general formula:

`<< r >> = a_0/(2Z)(3n^2 - l(l+1))`

`= a_0/(2 cdot 1) (3(2)^2 - 0(0+1)) = 6a_0` `color(blue)(sqrt"")`