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Physical Chemistry

Thomas Engel, Philip Reid

Chapter 20

The Hydrogen Atom - all with Video Answers

Educators

Chapter Questions

02:57

Problem 1

Calculate the wave number corresponding to the most and least energetic spectral lines in the Lyman, Balmer, and Paschen series for the hydrogen atom.

Shalini Tyagi
Shalini Tyagi
Numerade Educator
04:48

Problem 2

Show that the function $\left(r / a_{0}\right) e^{-r / 2 a_{0}}$ is a solution of the following differential equation for $l=1$
\[
\begin{aligned}
-\frac{\hbar^{2}}{2 m_{e} r^{2}} \frac{d}{d r}\left[r^{2} \frac{d R(r)}{d r}\right] & \\
&+\left[\frac{\hbar^{2} l(l+1)}{2 m_{e} r^{2}}-\frac{e^{2}}{4 \pi \varepsilon_{0} r}\right] R(r)=E R(r)
\end{aligned}
\]
What is the eigenvalue? Using this result, what is the value for the principal quantum number $n$ for this function?

Diogo Caetano
Diogo Caetano
Numerade Educator
04:47

Problem 3

Determine the probability of finding the electron in the region for which the $\psi_{320}$ wavefunction is negative (the toroidal region).

Sheh Lit Chang
Sheh Lit Chang
University of Washington
11:31

Problem 4

Calculate the expectation value for the potential energy of the $\mathrm{H}$ atom with the electron in the 1 s orbital. Compare your result with the total energy.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
View

Problem 5

Calculate the probability that the 1 sectron for $\mathrm{H}$ will be found between $r=a_{0}$ and $r=2 a_{0}$.

Victor Salazar
Victor Salazar
Numerade Educator
02:47

Problem 6

Calculate the distance from the nucleus for which the radial distribution function for the $2 p$ orbital has its main and subsidiary maxima.

Sai Chaitanya Tadepalli
Sai Chaitanya Tadepalli
Numerade Educator
02:50

Problem 7

Calculate the expectation value of the radius $\langle r\rangle$ at which you would find the electron if the $\mathrm{H}$ atom wave function is $\psi_{100}(r)$.

Bettina Hanlon
Bettina Hanlon
Numerade Educator
11:31

Problem 8

Calculate the expectation value for the kinetic energy of the $\mathrm{H}$ atom with the electron in the 2 s orbital. Compare your result with the total energy.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
06:17

Problem 9

Ions with a single electron such as $\mathrm{He}^{+}, \mathrm{Li}^{2+}$, and $\mathrm{Be}^{3+}$ are described by the $\mathrm{H}$ atom wave functions with $Z / a_{0}$ substituted for $1 / a_{0,}$ where $Z$ is the nuclear charge. The 1 s wave function becomes $\psi(r)=1 / \sqrt{\pi}\left(Z / a_{0}\right)^{3 / 2} e^{-Z r / a_{0}} .$ Using this result, calculate the total energy for the 1 state in $\mathrm{H}$, He $^{+}$, $\mathrm{Li}^{2}$, and $\mathrm{Be}^{3+}$ by substitution in the Schrödinger equation.

Zachary Warner
Zachary Warner
Numerade Educator
01:29

Problem 10

Ions with a single electron such as He $^{+}, \mathrm{Li}^{2+},$ and $\mathrm{Be}^{3+}$ are described by the $\mathrm{H}$ atom wave functions with $Z / a_{0}$ substituted for $1 / a_{0},$ where $Z$ is the nuclear charge. The 1 s wave function becomes $\psi(r)=1 / \sqrt{\pi}\left(Z / a_{0}\right)^{3 / 2} e^{-Z r / a_{0}},$ Using this result, compare the mean value of the radius $(r)$ at which you would find the 1 s electron in $\mathrm{H}, \mathrm{He}^{+}, \mathrm{L}_{1}^{2+}$, and $\mathrm{Be}^{3+}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:42

Problem 11

As the principal quantum number $n$ increases, the electron is more likely to be found far from the nucleus. It can be shown that for $\mathrm{H}$ and for ions with only one electron such
$\mathrm{He}^{+},\langle r\rangle_{n l}=\frac{n^{2} a_{0}}{Z}\left[1+\frac{1}{2}\left(1-\frac{l(l+1)}{n^{2}}\right)\right]$
Calculate the value of $n$ for an $s$ state in the hydrogen atom such that $\langle r\rangle=500 . a_{0} .$ Round up to the nearest integer. What is the ionization energy of the $\mathrm{H}$ atom in this state in electron-volts? Compare your answer with the ionization energy of the $\mathrm{H}$ atom in the ground state.

Rajesh Singh
Rajesh Singh
Numerade Educator
05:43

Problem 12

In this problem, you will calculate the probability of finding an electron within a sphere of radius $r$ for the H atom in its ground state.
a. Show using integration by parts, $\int u d \mathbf{v}=u v-\int v d u,$ that
\[
\int r^{2} e^{-r / \alpha} d r=e^{-r / \alpha}\left(-2 \alpha^{3}-2 \alpha^{2} r-\alpha r^{2}\right)
\]
b. Using this result, show that the probability of finding the electron within a sphere of radius $r$ for the hydrogen atom in its ground state is
\[
1-e^{-2 \gamma / a_{0}}-\frac{2 r}{a_{0}}\left(1+\frac{r}{a_{0}}\right) e^{-2 r / a_{0}}
\]
c. Evaluate this probability for $r=0.25 a_{0}, r=2.25 a_{0},$ and
\[
r=5.5 a_{0-}
\]

Suzanne W.
Suzanne W.
Numerade Educator
16:26

Problem 13

The radius of an atom $r_{\text {atomen }}$ can be defined as that value for which $90 \%$ of the electron charge is contained within a sphere of radius $r_{\text {atow}}$. Use the formula in $\mathrm{P} 20.12 \mathrm{b}$ to calculate the radius of the H atom.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
03:46

Problem 14

Use the result of $\mathrm{P} 20.13$.
a. Calculate the mass density of the H atom.
b. Compare your answer with the nuclear density assuming a nuclear radius of $1.0 \times 10^{-15} \mathrm{m}$
c. Calculate the mass density of the H atom outside of the nucleus.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:21

Problem 15

Calculate the expectation value $\langle r-\langle r\rangle\rangle^{2}$ if the H atom wave function is $\psi_{100}(r)$.

Timothy James
Timothy James
Numerade Educator
03:32

Problem 16

In spherical coordinates, $z=r \cos \theta$. Calculate $\langle z\rangle$ and $\left\langle z^{2}\right\rangle$ for the $H$ atom in its ground state. Without doing the calculation, what would you expect for $\langle x\rangle$ and $\langle y\rangle,$ and $\left\langle x^{2}\right\rangle$ and $\left(y^{2}\right\rangle ?$ Why?

Chris Trentman
Chris Trentman
Numerade Educator
01:04

Problem 17

The force acting between the electron and the proton in the $\mathrm{H}$ atom is given by $F=-e^{2} / 4 \pi \varepsilon_{0} r^{2}$. Calculate the expectation value $\langle F\rangle$ for the 1 and $2 p_{z}$ states of the $\mathrm{H}$ atom in terms of $e, \varepsilon_{0},$ and $a_{0}$.

Raj Bala
Raj Bala
Numerade Educator
01:09

Problem 18

The d orbitals have the nomenclature $d_{z}^{2}, d_{x y}, d_{x z}$ $d_{y_{\Sigma}}$ and $d_{x^{2}-y^{2}}$ Show how the $d$ orbital
\[
\psi_{3, c}(r, \theta, \phi)=\frac{\sqrt{2}}{81 \sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} \frac{r^{2}}{a_{0}^{2}} e^{-r / 3 a_{0}} \sin \theta \cos \theta \sin \phi
\]
can be written in the form $y z F(r)$

Hitendra Singh
Hitendra Singh
Numerade Educator
04:34

Problem 19

Calculate the expectation value of the moment of inertia of the $\mathrm{H}$ atom in the $2 \mathrm{s}$ and $2 p_{z}$ states in terms of $\mu$ and $a_{0}$.

Pawan Yadav
Pawan Yadav
Numerade Educator
02:59

Problem 20

The energy levels for ions with a single electron such as $\mathrm{He}^{+}, \mathrm{Li}^{2+},$ and $\mathrm{Be}^{3+}$ are given by $E_{n}=-Z^{2} e^{2} /\left(8 \pi \varepsilon_{0} a_{0} n^{2}\right), n=1,2,3,4, \ldots . .$ Calculate the ionization energies of $\mathrm{H}, \mathrm{He}^{+}, \mathrm{Li}^{2+},$ and $\mathrm{Be}^{3+}$ in their ground states in units of electron-volts (eV).

Adriano Chikande
Adriano Chikande
Numerade Educator
03:58

Problem 21

Calculate the mean value of the radius $\langle r\rangle$ at which you would find the electron if the $\mathrm{H}$ atom wave function is $\psi_{210}(r, \theta, \phi)$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:48

Problem 22

The total energy eigenvalues for the hydrogen atom are given by $E_{n}=-e^{2} /\left(8 \pi \varepsilon_{0} a_{0} n^{2}\right), n=1,2,3,4, \dots,$ and
the three quantum numbers associated with the total energy eigenfunctions are related by $n=1,2,3,4, \ldots ; l=0,1,2,3$ $\ldots, n-1 ;$ and $m_{l}=0,\pm 1,\pm 2,\pm 3, \ldots \pm l$
Using the nomenclature $\psi_{n l m_{l}}$ list all eigenfunctions that have the following total energy eigenvalues:
a. $E=-\frac{e^{2}}{32 \pi \varepsilon_{0} a_{0}}$
b. $E=-\frac{e^{2}}{72 \pi \varepsilon_{0} a_{0}}$
$\mathbf{c}, E=-\frac{e^{2}}{128 \pi \varepsilon_{0} a_{0}}$
d. What is the degeneracy of each of these energy levels?

Linda Winkler
Linda Winkler
Numerade Educator
04:38

Problem 23

Locate the radial and angular nodes in the H orbitals $\psi_{3 p_{x}}(r, \theta, \phi)$ and $\psi_{3 p_{t}}(r, \theta, \phi)$

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
05:06

Problem 24

Calculate the average value of the kinetic and potential energies for the $\mathrm{H}$ atom in its ground state.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:31

Problem 25

Show by substitution that $\psi_{100}(r, \theta, \phi)=$ $1 / \sqrt{\pi}\left(1 / a_{0}\right)^{3 / 2} e^{-r / a_{0}}$ is a solution of
\[
\begin{array}{r}
-\frac{\hbar^{2}}{2 m_{e}}\left[\begin{array}{c}
\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi(r, \theta, \phi)}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi(r, \theta, \phi)}{\partial \theta}\right) \\
+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \psi(r, \theta, \phi)}{\partial \phi^{2}}
\end{array}\right. \\
-\frac{e^{2}}{4 \pi \varepsilon_{0} r} \psi(r, \theta, \phi)=E \psi(r, \theta, \phi)
\end{array}
\]
What is the eigenvalue for the total energy? Use the relation $a_{0}=\varepsilon_{0} h^{2} /\left(\pi m_{c} e^{2}\right)$ to simplify your answer.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:40

Problem 26

Show that the total energy eigenfunctions $\psi_{100}(r)$ and $\psi_{200}(r)$ are orthogonal.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
08:44

Problem 27

As will be discussed in Chapter 21 , core electrons shield valence electrons so that they experience an effective nuclear charge $Z_{c f}$ rather than the full nuclear charge. Given that the first ionization energy of $\mathrm{Li}$ is $5.39 \mathrm{eV}$, use the for mula in Problem $\mathrm{P} 20.20$ to estimate the effective nuclear charge experienced by the 2 s electron in $\mathrm{Li}$.

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
01:11

Problem 28

Is the total energy wave function
\[
\psi_{310}(r, \theta, \phi)=\frac{1}{81}\left(\frac{2}{\pi}\right)^{1 / 2}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(6 \frac{r}{a_{0}}-\frac{r^{2}}{a_{0}^{2}}\right) e^{-r / 3 a_{0}} \cos \theta
\]
an eigenfunction of any other operators? If so, which ones? What are the eigenvalues?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:48

Problem 29

Show that the total energy eigenfunctions $\psi_{210}(r, \theta, \phi)$ and $\psi_{211}(r, \theta, \phi)$ are orthogonal. Do you have to integrate over all three variables to show that the functions are orthogonal?

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
06:34

Problem 30

Calculate $\langle r\rangle$ and the most probable value of $r$ for the $\mathrm{H}$ atom in its ground state. Explain why they differ with a drawing.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:55

Problem 31

How many radial and angular nodes are there in the following H orbitals?
a. $\psi_{2 p_{k}}(r, \theta, \phi)$
b. $\psi_{2}(r)$
$\mathbf{c} . \psi_{3 \alpha_{x_{k}}}(r, \theta, \phi)$
d. $\psi_{3 d_{x^{2}-\gamma^{2}}}(r, \theta, \phi)$

Shalini Tyagi
Shalini Tyagi
Numerade Educator
04:55

Problem 32

Show that $\psi_{2 p_{x}}(r, \theta, \phi)$ and $\psi_{2 p_{y}}(r, \theta, \phi)$ can be written in the form $N x e^{-r / 2 a_{0}}$ and $N^{\prime} y e^{-r / 2 a_{0}}$ where $N$ and $N^{\prime}$ are normalization constants.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:47

Problem 33

Using the result of Problem $\mathrm{P} 20.12$, calculate the probability of finding the electron in the 1 state outside a sphere of radius $0.75 a_{0}, 2.5 a_{0},$ and $4.5 a_{0}$

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator