Question 42.6: Radial Probability density for Hydrogen in 2p states Figure ......
Radial Probability density for Hydrogen in 2p states
Figure 42.16 shows the radial probability density for hydrogen in both the n = 2, \ell = 0 and n = 2, \ell = 1 subshells. The radial probability density for the \ell = 1 subshell peaks at r_{max} = 4r_B. The radial probability density for the \ell = 0 peaks at a greater distance; find this r_{max} in terms of the Bohr radius r_B.
INTERPRET and ANTICIPATE
We know the wave function for this state. To find the radial probability density P(r), we’ll need to square this wave function and multiply the result by 4\pi r^2 . To find the maximum, we use the fact that the derivative of a function is zero at its extrema.
SOLVE
We substitute the wave function (Eq. 42.21)
into Equation 42.19 for the radial probability density.
\begin{aligned}& P(r)=4 \pi r^2|\psi|^2 \quad \quad (42.19)\\& P(r)=4 \pi r^2\left[\frac{1}{\sqrt{2 \pi r_{ B }^3}}\left(\frac{1}{2}-\frac{r}{4 r_{ B }}\right) e^{-r / 2 r_{ B }}\right]^2\end{aligned}We need a few steps of algebra. Because we will be taking the derivative and setting it to zero, we can make the algebra a little easier if we replace the equals with a proportionality and drop the leading constants.
Now take the derivative with respect to r and set the derivative to zero.
\begin{aligned}& \frac{d P}{d r} \propto 2\left(2 r-\frac{r^2}{r_{ B }}\right)\left(2-\frac{2 r}{r_{ B }}\right) e^{-r / r_{ B }}-\frac{1}{r_{ B }}\left(2 r-\frac{r^2}{r_{ B }}\right)^2 e^{-r / r_{ B }}=0 \\& 2\left(2-\frac{2 r}{r_{ B }}\right)-\frac{1}{r_{ B }}\left(2 r-\frac{r^2}{r_{ B }}\right)=0 \\& 4-\frac{4 r}{r_{ B }}-\frac{2 r}{r_{ B }}+\frac{r^2}{r_{ B }^2}=0 \\& 4-\frac{6 r}{r_{ B }}+\frac{r^2}{r_{ B }^2}=0\end{aligned}
Finally, use the quadratic formula to solve for r in terms of the Bohr radius r_B. We choose the + sign. The – sign gives the smaller peak shown in Figure 42.16.
r=\frac{6 r_{ B } \pm \sqrt{36 r_{ B }^2-4\left(4 r_{ B }^2\right)}}{2}=5.24 r_{ B }CHECK AND THINK
Compare the answer here to the ground-state level of hydrogen (Fig. 42.13). It makes sense that when hydrogen is in a higher energy state, the electron is most likely to be found farther from the nucleus.
