Class 2
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Bohr Model of the Hydrogen Atom
The Bohr Model of the hydrogen atom assumes that the atom consists of one electron
“orbiting” a positively charged nucleus. Although it does NOT do a good job of
describing atoms that have more than one electron (which most atoms do), it is still
helpful for learning basic concepts. It helps us to understand the origin (direction) and
the magnitude (size) of the interactive forces between a negatively charged electron and
the positively charged nucleus. The Bohr model is described in Lesson 2 of the Chem
110 e-book.
We have three learning goals for this Module.
1. Understand the charge and distance dependence of charge interactions. You will
be able to identify whether the electron is easier or harder to remove when the
distance or magnitude of charge is varied. You will compute the force of
attraction between two charges using Coulomb’s law. You will recognize that a
negative value for force is attraction, and a positive value is repulsion. You will
be able to construct a plot of the relationship between energy and distance
dependence of the orbits.
2. Know how to draw and interpret energy level diagrams for the Bohr model of the
hydrogen atom. You will construct an energy level diagram for the Bohr model
and correctly illustrate the intervals between energy levels.
3. Understand what is meant by the energy of an electron. You will be able to
describe what is meant by the energy of an electron. You will compute the
allowed energy states for the electron given by E = RH (1/n2). You will be able to
calculate the energy of the hydrogen atom for different values of n in terms of RH
and recognize which orbit has the highest energy. You will recognize that the
energy of the electron at n = ∞ is zero.
Pre-Requisites and Review
In order to be successful at these Module activities, there are several things you must
already be able to do. Lessons referred to are the section in the Chem 110 e-book.
1. Be able to convert between metric units. Use ALEKs to review this topic.
2. Describe the basic structure of the atom, including the number and location of
neutrons, protons and electrons. (Lesson 1, Unit 1.)
3. Be able to solve equations, work with formulas and significant figures, and
demonstrate knowledge of dimensional analysis. Know the difference between
absolute value and magnitude. (Review this material in ALEKs.)
4. Recall the definition of an electron, proton and neutron, and state their charges
and relative mass. (Review Lesson 1, unit 1.)
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Bohr Activity 1: Energy vs. Distance
1A. This first activity studies the effect of distance on the interaction of charged
particles. Do the following pairs of charged particles attract or repel each other?
Circle the correct choice.
+1
+1
i.
-1
+1
repel
attract
repel
attract
repel
-1
ii.
iii.
attract
-1
1B. Summarize what this says about how charged particles interact;
1C. Coulomb’s Law is described in the text in sections 2.3 and 5.1; it tells us how the
energy of interaction (E) is related to charge and distance. Q1, and Q2 are the
charges on the particles, and d is the distance between the centers of the particles.
We will assume that the constant k is 1 to simplify our calculations. The quantities
Q1, Q2 and d are labeled on the following picture.
E=
kQ1Q2
d
Q1 = +1
-1
+1
d
Q2 = −1
Label the following picture with Q1, Q2 and d.
+1
Q1 =
Q2 =
+2
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Bohr Activity 1: continued
1D. Decide whether each picture shows attractive or repulsive energy.
What are Q1 and Q2?
i. Write your answer, then draw a line on each picture and label “d” as in 1C above:
Figure a.
Figure b.
+1
+1
-1
-1
_________________
E=
kQ1Q2
d
_________________
Figure c.
Figure d.
-1
+1
+1
-1
_________________
_________________
ii. In which picture are the charges closest together?
iii. In which picture is the interaction the strongest? (Use the equation above.)
1E. Determine which interacting pair would be the hardest to pull apart:
Fig. a or b?
Fig. a or c?
Fig. b or c?
Fig. c or d?
1
1F. Notice that the energy is proportional to in Coulomb’s Law. Place the four
d
pictures above in order of increasing magnitude of energy, using Coulomb’s Law to
evaluate qualitatively.
__________ < __________ < ___________ < __________
1G. Now, summarize what you know.
i. Opposite charges __________________________.
ii. Like charges ______________________________.
iii. As two opposite charges get closer together, the magnitude of the attractive
energy _____________________.
iv. As two like charges get closer together, the magnitude of the repulsive energy
_____________________.
YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. ________
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Bohr Activity 2: Energy vs. Magnitude of Charge
2A. Assume d is the same for all of the pictures below. Decide whether each picture
shows attractive or repulsive energy:
Figure a.
-1
Figure b.
+1
_______________
+2
-1
+2
E=
kQ1Q2
d
-3
-2
Figure c.
_________________
-1
Figure d.
______________
_________________
2B. Determine which interacting pair would be the hardest to pull apart:
Fig. a or b?
Fig. a or d?
Fig. c or d?
Fig. b or c?
Fig. a or c?
2C. Use “less than” symbols to place the four pictures in order of increasing overall
energy, using Coulomb’s Law above qualitatively. Negative energy values are less
than positive energy values.
2D. Now, summarize what you know.
i. As charge increases, the magnitude of the attractive energy _________________.
ii. As charge decreases, the magnitude of the attractive energy _________________.
What happens to the magnitude of the repulsive energy between two like charges when
the charges increase?
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Bohr Activity 3: Calculations with Coulomb’s Law
3A. Use the picture below for this activity; is the energy attractive or repulsive?
3B. Label Q1, Q2, and d in the picture.
3C. To simplify our calculations, we won’t worry about units yet, and we will assume
that k=1. Write the value for each variable below. What is the energy of the
interaction (E) equal to?
kQ1Q2
E=
d
Q1 =
+1
-1
1
2
Q2 =
d=
0
k=
Distance
E=
-----------------------------------------------------------------------------------------------------------3D. Use Coulomb’s Law to answer the following questions:
E=
kQ1Q2
d
(Assume k=1 to simplify.)
i. Calculate the energy E.
+1
Is the interaction attractive or repulsive?
+1
0
1
2
3
Distance
Angstroms
------------------------------------------------------------------------------------------------------------
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Bohr Activity 3: continued
-----------------------------------------------------------------------------------------------------------ii. Calculate the energy.
0
Is the interaction attractive or repulsive?
-1
-1
1
2
3
Distance
Angstroms
-----------------------------------------------------------------------------------------------------------iii. Calculate the energy.
+2
Is the interaction attractive or repulsive?
-1
0
1
2
3
Distance
Angstroms
-----------------------------------------------------------------------------------------------------------iv. Calculate the energy.
+2
Is the interaction attractive or repulsive?
-2
0
1
2
3
Distance
-----------------------------------------------------------------------------------------------------------Angstroms
3E. Now, summarize what you know.
i. Energy is negative when the charges are _______________________.
ii. Energy is positive when the charges are _______________________.
iii. Energy is negative when the interaction is ______________________.
iv. Energy is positive when the interaction is ______________________.
The sign of the Energy of interaction provides
information about whether it is attractive or
repulsive.
The absolute value of the Energy of interaction
(the magnitude) provides information about its
relative strength.
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Bohr Activity 4: Calculating Energy vs. Distance
4A. Use Coulomb’s Law to calculate the energy of the following interactions. Assume
that k=1, and
i.
+1
E=
kQ1Q2
. Please write out expressions completely!
d
E=
−1
d=1
ii.
+1
E=
−1
d=2
iii.
+1
E=
−1
d=4
iv. +1
−1
d=8
E=
v. Are these interactions attractive or repulsive?
vi. What is the sign of the interactive energy in each case?
vii. What happens to the numerator in the calculations of Coulomb’s Law as distance
increases?
viii. What happens to the denominator in the calculations of Coulomb’s Law as distance
increases?
ix. Now look at the ratio of the numerator to the denominator as the distance gets larger.
What happens to the magnitude of the energy E as distance increases? (Focus on the
absolute value, not the sign of the energy.)
YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. _______
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Bohr Activity 4: Continued
4B. Fill in the following table using the values calculated in 4A.
Distance (d)
Energy (E)
4C. Using the data from the table above, make a plot of the data, and draw a smooth
curve to connect the points. Use a pencil.
Energy vs. Distance for Charge Interaction
0
-0.1
-0.2
Energy (Joules)
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2
0
1
2
3
4
5
6
Distance (units)
7
8
9
10
4D. At which distance is the interaction the strongest? (Note that all the energy values
are negative.)
4E. What does the graph tell you about the relationship between distance and the
magnitude of the energy?
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Bohr Activity 4: Continued
Notice that this energy E is the same as the interaction
energy you worked with in previous activities. Here
the comparison is quantitative (you worked with
actual numbers and data to reach a conclusion
about the quantity of energy.) In previous activities
the comparison was qualitative (you used logic and
knowledge to reach a conclusion about the quality
of the energy of interaction.) Note that you can use
qualitative reasoning to check your quantitative
answers.
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Bohr Activity 4: continued
4F. An energy level diagram is a way of comparing energy differences between different
situations.
i. For each of the four interactions in 4A, draw a line next to its corresponding
energy on the energy level diagram below. Label each line with the
corresponding distance between the charges, d. The line corresponding to the
interaction “d=8” is drawn for you.
0
-0.10
Draw lines like
this one.
+1
-1
d=8
Energy
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
ii. To the right of each line, draw the corresponding charge interaction, using a double
headed arrow, estimating d, and keeping each distance approximately correct. The
d=8 interaction is drawn for you.
iii. Which interaction has the strongest interaction (highest magnitude energy)?
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Bohr Activity 4: continued
i. Which interaction has the lowest overall energy? (Note that a zero value is the
highest energy on the above diagram)
ii. As the interaction between oppositely charged particles gets stronger, they are
harder to pull apart. Name two ways to make the interaction get stronger (review
Activities 2 and 3 and look at the relationships in Coulomb's Law).
iii. With a stronger interaction, the system is more stable, and harder to perturb (i.e.
by pulling apart.) With a stronger, more stable interaction, the magnitude of the
energy is _____________________.
iv. Even though the magnitude of the energy is greater for a strong interaction, the
sign of the energy is ___________________ (recall that this is an attractive
interaction).
Therefore, the stronger interaction must have a lower overall energy (large negative
numbers are smaller than small negative numbers.)
Recall that the interactive energy E is also called the attractive or repulsive energy.
The stronger the interaction between
oppositely charged particles, the more
stable the system, and the lower the
overall energy.
Notice that the strongest interaction is of lowest overall energy
even though the magnitude is greater.
Also notice that if the distance increases enough (such as to
infinity), the energy of interaction will be zero and there will be
no interaction at all.
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Bohr Activity 5: Calculating the Energy of Orbits
The Bohr model of the hydrogen atom is described in Lesson 2.1 in the e-book. The
figures below show the model, in which a hydrogen atom is one proton (+) with an
electron (−) “orbiting” around it. There are many possible “orbits”. We label the
closest orbit as n=1, the next as n=2, etc.
kQ Q
5A. Calculate E, assuming k=1, d=n2 and E = 1 2 . i is done for you.
d
etc. . . .
i. Electron in the n=1 orbit.
Q1 = +1 (proton)
Q2 = -1 (electron)
d = n2 = (1)2
(1)(+1)(−1) = −1
So, E =
n=4
n=3
1
n=1
n=2
------------------------------------------ii. Electron in the n=2 orbit.
etc. . . .
E=
n=1
n=2
n=4
n=3
------------------------------------------------------------------------------------------------iii. Electron in the n=3 orbit.
etc. . . .
E=
n=1
n=2
n=3
n=4
--------------------------------------------------------------------iv. Electron in the n=4 orbit.
etc. . . .
E=
n=1
n=2
n=4
n=3
-------------------------------------------------------------------------------------------------
---
YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. ______
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Bohr Activity 5: continued
5B. Use the data from part A to fill in the table for each value of n in the chart below.
5C. Using the data from part B, plot the interactive energy of the electron vs. distance
from the nucleus. Recall that you must draw a smooth curve.
Energy vs. Distance for Charge Interaction
0
n= d= E=
2
3
4
-0.3
Energy (Joules)
1
-0.1
-0.2
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
Distance (units)
Because the sign of the energy is negative, we know that the energy is attractive. Now
focus on the magnitude of the energy. What does the graph tell you about the
relationship between the orbit number n, the distance from the nucleus, and the
magnitude of the energy?
When the distance between the particles
decreases, the attractive interaction increases, the
sign of the interactive energy is negative, the
magnitude of the energy increases, and the
overall energy decreases.
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Bohr Activity 6: Energy Level Diagram
6A. Now construct a traditional energy level diagram for the Bohr model. To simplify
we will assume k=1 and d = n2. Use the numbers from the table you completed in
Activity 5B. For each value of n, draw a line next to its corresponding energy on
the scale below. Label each line with the value of n it represents. The line
corresponding to n = 1 is drawn for you.
Figure 1. Energy level diagram for k=1
E=
kQ1Q2
d
0
-0.10
Energy
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
n=1
-1.00
6B. What do you notice about the distance between the energy levels as n increases?
6C. As the value of n increases, the magnitude of the energy ________________.
6D. As the value of n increases, the overall energy of the system______________.
6E. If the electron is completely removed from the system (n = ∞), what would the
overall energy of the system be? (Hint; Look at Activity 4F)
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Bohr Activity 6: continued.
6F. Compare your figure in 6A (Figure 1) to Figure 2 below. (Compare this to Figure F022-1 in the ebook). The lines you drew represent the allowed energy levels for the
electron. The arrows on the diagram correspond to electronic transitions, which are
covered in Activity 8. If you continued to add energy levels for greater values of n to
figure 1, your diagram would look more like Figure 2.
Notice that if k=hcRH , the diagrams are the same.
Each energy level on Figure 1 is multiplied by k, which we assumed was 1 in
order to simplify our calculations. Each energy level on the diagram in Figure 2 is
multiplied by the constant hcRH. Here h is Planck’s constant, c is the speed of light, and
RH is the Rydberg constant.
Given the following information, what is the value of the constant “hcRH”? Be careful
with canceling the units.
h = 6.63 x 10-34 Js
c = 3.00 x 108 ms-1
RH = 1.096776 x 107 m-1
n=4
−1/9 hcRH
n=3
−1/4 hcRH
n=2
n=5
Energy
k = hcRH =
0
−1/16hcRH
Figure 2. Bohr model energy level
diagram for k=hcRH
−hcRH
n=1
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Bohr Activity 6: continued.
6G. Use the previous activities to answer the following questions:
i.
Which electron has a greater magnitude of energy (stronger interactive
energy); n = 1 or n = 2? (Look back at 6C.)
ii.
Which electron has greater overall energy; n = 1 or n = 2? (Look at Figure 1
in 6A.)
iii.
Which electron is harder to remove from its orbit; n = 1 or n = 2? How do
you know this?
iv.
Which electron has greater overall energy: n = 1 or n = 5?
Note that general chemistry questions about the energy
of an electron usually refer to the overall interactive
energy, not the magnitude. The lower the energy, the
more negative the energy, and the more stable the state.
6H. Up to this point, we have assumed that k=1 to simplify our calculations. Now,
assume that k = hcRH, which is the number calculated in 6D. Also assume that d =
n2, which is a good approximation. Write out the equation that you would solve,
then calculate a numerical value for E.; what is the overall energy in Joules of an
electron in the n = 1 level? (Use Coulomb’s Law for the calculation.)
6I. Substitute the following values into Coulomb’s Law, which is
E=
kQ1Q2
and
d
simplify to obtain a general formula for E. This E is what the energy of the electron in
a hydrogen atom is equal to, and can be used for to find the energy for any value of n.
Q1 = −1
Q2 = +1
k = hcRH
d = n2
E=
This general equation is used to calculate the energy
of an electron in any orbit in the H-atom. It is
equation E02-2-3 in the e-book, with Z = +1.
YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. _______
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Bohr Activity 7: Energy of an Electron
Note that adding energy to a system (absorbing
energy) results in raising its overall energy level.
Removing energy from a system (emitting energy)
results in lowering its overall energy level.
Use Figure 3 to answer the following questions.
Figure 3. Orbits of the electron in a hydrogen atom.
etc. . . .
n=1
n=2
n=3
n=4
7A. The “potential energy of an electron” is related to the interaction between the
electron and the protons in the nucleus. Is the interaction attractive or repulsive?
7B. What is the sign of the potential energy of the electron in any orbit?
7C. Taking the electron completely away from the nucleus is like moving it to the n=∞
level on Figure 2. Compare the energy of n=1 to n=∞. We must add energy to the
system to get the electron to be in a higher energy level (with higher n value.) So, to
completely remove an electron from the n = 1 orbit, do you put energy in or get
energy out of the system?
7D. Do you put energy in or get energy out of the system to place an electron into an
orbit?
7E. Choose the correct definition of the “energy of the electron”. (Remember that the
energy of the electron is negative. (= −hcRH/n2)
a) The energy of an electron is the energy given off to remove an electron from its
“orbit”.
b) The energy of an electron is the energy given off when an electron is placed
into its “orbit”.
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