Question

What is the probability that in the ground state of hydrogen atom , the electron will be found at a radius greater than the Bohr radius?

Short answer

The probability is P = 68% .

Step-by-step solution

Identification of the given data:

The given data is listed below.

The radius of the electron is greater than the Bohr radius.

Formula for finding the probability of electron:

The formula for finding the probability of electron in the ground state of hydrogen atom inside a sphere of radius r is given by,

$\mathbf{p}\left(r\right)\mathbf{=}\mathbf{1}\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{a}}\left(1+2x+2x^2\right)$

Here, x = 1 and r = a .

Here, a is the Bohr radius.

Determine the probability of the electron of the hydrogen atom in its ground state:

The probability of finding the electron in the ground state of a hydrogen atom found inside a sphere of radius r is given by-

$P\left(r\right)=\left[1-e^{-2x}\left(1+2x+2x^2\right)\right]$

Here, x = na and a is the Bohr radius.

For, r = a and x = 1 .

$$\begin{gathered} P\left(a\right)=\left[1-e^{-2}\left(1+2+2\right)\right] \\ =1-5e^{-2} \\ =1-\left(5\times 0.135\right) \\ =0.323 \end{gathered}$$

Now, the probability that the electron can be found outside this sphere is:

$$\begin{gathered} P=1-0.322 \\ =0.677 \end{gathered}$$

$$\begin{gathered} P\%=\left(0.677\times 100\right)\% \\ =68\% \end{gathered}$$

Thus, the probability the electron will be found at a radius greater than the Bohr radius is 68% .