Class 2 page 1 Bohr Moel of the Hyrogen Atom The Bohr Moel of the hyrogen atom assumes that the atom consists of one electron orbiting a positively charge nucleus. Although it oes NOT o a goo job of escribing atoms that have more than one electron (which most atoms o), it is still helpful for learning basic concepts. It helps us to unerstan the origin (irection) an the magnitue (size) of the interactive forces between a negatively charge electron an the positively charge nucleus. The Bohr moel is escribe in Lesson 2 of the Chem 110 e-book. We have three learning goals for this Moule. 1. Unerstan the charge an istance epenence of charge interactions. You will be able to ientify whether the electron is easier or harer to remove when the istance or magnitue of charge is varie. 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, an a positive value is repulsion. You will be able to construct a plot of the relationship between energy an istance epenence of the orbits. 2. Know how to raw an interpret energy level iagrams for the Bohr moel of the hyrogen atom. You will construct an energy level iagram for the Bohr moel an correctly illustrate the intervals between energy levels. 3. Unerstan what is meant by the energy of an electron. You will be able to escribe what is meant by the energy of an electron. You will compute the allowe energy states for the electron given by E = R H (1/n 2 ). You will be able to calculate the energy of the hyrogen atom for ifferent values of n in terms of R H an recognize which orbit has the highest energy. You will recognize that the energy of the electron at n = is zero. Pre-Requisites an Review In orer to be successful at these Moule activities, there are several things you must alreay be able to o. Lessons referre 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, incluing the number an location of neutrons, protons an electrons. (Lesson 1.) 3. Be able to solve equations, work with formulas an significant figures, an emonstrate knowlege of imensional analysis. Know the ifference between absolute value an magnitue. (Review this material in ALEKs.) 4. Recall the efinition of an electron, proton an neutron, an state their charges an relative mass. (Review Lesson 1.) Spring 2014 Page 2 Group Work
Class 2 page 2 Bohr Activity 1: Energy vs. Distance 1A. This first activity stuies the effect of istance on the interaction of charge particles. Do the following pairs of charge particles attract or repel each other? Circle the correct choice. i. attract repel ii. -1-1 attract repel iii. -1 attract repel 1B. Summarize what this says about how charge particles interact; 1C. Coulomb s Law is escribe in the text in sections 2.3 an 5.1; it tells us how the energy of interaction (E) is relate to charge an istance. Q 1, an Q 2 are the charges on the particles, an is the istance between the centers of the particles. We will assume that the constant k is 1 to simplify our calculations. The quantities Q 1, Q 2 an are labele on the following picture. E kq 1Q 2 Q 1 = -1 Q 2 = 1 Label the following picture with Q 1, Q 2 an. Q 1 = Q 2 = +2 Spring 2014 Page 3 Group Work
Class 2 page 3 Bohr Activity 1: continue 1D. Decie whether each picture shows attractive or repulsive energy. What are Q 1 an Q 2? i. Write your answer, then raw a line on each picture an label as in 1C above: Figure a. Figure b. -1-1 E kq 1Q 2 Figure c. Figure. -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 woul be the harest to pull apart: Fig. a or b? Fig. a or c? Fig. b or c? Fig. c or? 1F. Notice that the energy is proportional to 1 in Coulomb s Law. Place the four pictures above in orer of increasing magnitue 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 magnitue of the attractive energy. iv. As two like charges get closer together, the magnitue of the repulsive energy. YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. Spring 2014 Page 4 Group Work
Class 2 page 4 Bohr Activity 2: Energy vs. Magnitue of Charge 2A. Assume is the same for all of the pictures below. Decie whether each picture shows attractive or repulsive energy: Figure a. Figure b. -1 +2-1 E kq 1Q 2 Figure c. Figure. +2-2 -3-1 2B. Determine which interacting pair woul be the harest to pull apart: Fig. a or b? Fig. a or? Fig. c or? Fig. b or c? Fig. a or c? 2C. Use less than symbols to place the four pictures in orer 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 magnitue of the attractive energy. ii. As charge ecreases, the magnitue of the attractive energy. What happens to the magnitue of the repulsive energy between two like charges when the charges increase? Spring 2014 Page 5 Group Work
Class 2 page 5 Bohr Activity 3: Calculations with Coulomb s Law 3A. Use the picture below for this activity; is the energy attractive or repulsive? 3B. Label Q 1, Q 2, an in the picture. 3C. To simplify our calculations, we won t worry about units yet, an we will assume that k=1. Write the value for each variable below. What is the energy of the interaction (E) equal to? E kq 1 Q 2 Q 1 = -1 Q 2 = = k = 0 1 2 3 Distance E = ------------------------------------------------------------------------------------------------------------ 3D. Use Coulomb s Law to answer the following questions: E kq 1Q 2 (Assume k=1 to simplify.) i. Calculate the energy E. Is the interaction attractive or repulsive? 0 1 2 3 Distance ------------------------------------------------------------------------------------------------------------ Spring 2014 Page 6 Group Work
Class 2 page 6 Bohr Activity 3: continue ------------------------------------------------------------------------------------------------------------ ii. Calculate the energy. -1-1 0 1 2 3 Is the interaction attractive or repulsive? Distance ------------------------------------------------------------------------------------------------------------ iii. Calculate the energy. +2-1 Is the interaction attractive or repulsive? 0 1 2 3 Distance ------------------------------------------------------------------------------------------------------------ iv. Calculate the energy. +2-2 Is the interaction attractive or repulsive? 0 1 2 3 Distance ------------------------------------------------------------------------------------------------------------ 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 provies information about whether it is attractive or repulsive. The absolute value of the Energy of interaction (the magnitue) provies information about its relative strength. Spring 2014 Page 7 Group Work
Class 2 page 7 Bohr Activity 4: Calculating Energy vs. Distance 4A. Use Coulomb s Law to calculate the energy of the following interactions. Assume i. that k=1, an E kq 1Q 2. Please write out expressions completely! 1 E = = 1 ii. 1 E = = 2 iii. 1 E = = 4 iv. 1 = 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 istance increases? viii. What happens to the enominator in the calculations of Coulomb s Law as istance increases? ix. Now look at the ratio of the numerator to the enominator as the istance gets larger. What happens to the magnitue of the energy E as istance increases? (Focus on the absolute value, not the sign of the energy.) YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. Spring 2014 Page 8 Group Work
Class 2 page 8 Bohr Activity 4: Continue 4B. Fill in the following table using the values calculate in 4A. Distance () Energy (E) 4C. Using the ata from the table above, make a plot of the ata, an raw a smooth curve to connect the points. Use a pencil. Energy vs. Distance for Charge Interaction 0-0.1-0.2-0.3 Energy (Joules) -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 Distance (units) 4D. At which istance is the interaction the strongest? (Note that all the energy values are negative.) 4E. What oes the graph tell you about the relationship between istance an the magnitue of the energy? Spring 2014 Page 9 Group Work
Class 2 page 9 Bohr Activity 4: Continue Notice that this energy E is the same as the interaction energy you worke with in previous activities. Here the comparison is quantitative (you worke with actual numbers an ata to reach a conclusion about the quantity of energy.) In previous activities the comparison was qualitative (you use logic an knowlege to reach a conclusion about the quality of the energy of interaction.) Note that you can use qualitative reasoning to check your quantitative answers. Spring 2014 Page 10 Group Work
Class 2 page 10 Bohr Activity 4: continue 4F. An energy level iagram is a way of comparing energy ifferences between ifferent situations. i. For each of the four interactions in 4A, raw a line next to its corresponing energy on the energy level iagram below. Label each line with the corresponing istance between the charges,. The line corresponing to the interaction =8 is rawn for you. 0-0.10-0.20 Draw lines like this one. =8-1 Energy -0.30-0.40-0.50-0.60-0.70-0.80-0.90-1.00 ii. To the right of each line, raw the corresponing charge interaction, using a ouble heae arrow, estimating, an keeping each istance approximately correct. The =8 interaction is rawn for you. iii. Which interaction has the strongest interaction (highest magnitue energy)? Spring 2014 Page 11 Group Work
Class 2 page 11 Bohr Activity 4: continue i. Which interaction has the lowest overall energy? (Note that a zero value is the highest energy on the above iagram) ii. As the interaction between oppositely charge particles gets stronger, they are harer to pull apart. Name two ways to make the interaction get stronger (review Activities 2 an 3 an look at the relationships in Coulomb's Law). iii. With a stronger interaction, the system is more stable, an harer to perturb (i.e. by pulling apart.) With a stronger, more stable interaction, the magnitue of the energy is. iv. Even though the magnitue 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 calle the attractive or repulsive energy. The stronger the interaction between oppositely charge particles, the more stable the system, an the lower the overall energy. Notice that the strongest interaction is of lowest overall energy even though the magnitue is greater. Also notice that if the istance increases enough (such as to infinity), the energy of interaction will be zero an there will be no interaction at all. Spring 2014 Page 12 Group Work
Class 2 page 12 Bohr Activity 5: Calculating the Energy of Orbits The Bohr moel of the hyrogen atom is escribe in section 6.3 in the text. The figures below show the moel, in which a hyrogen atom is one proton (+) with an electron ( ) orbiting aroun it. There are many possible orbits. We label the closest orbit as n=1, the next as n=2, etc. 5A. Calculate E, assuming k=1, =n 2 an E kq Q 1 2. i is one for you. i. Electron in the n=1 orbit. etc... Q 1 = (proton) Q 2 = -1 (electron) = n 2 = (1) 2 1 So, E 1 1 1 1 n = 1 n = 2 n = 3 n = 4 ----------------------------------------------------------------------------------------------------------- ii. Electron in the n=2 orbit. etc... E = n = 1 n = 2 n = 3 n = 4 ----------------------------------------------------------------------------------------------------------- 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 = 3 n = 4 ----------------------------------------------------------------------------------------------------------- YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. Spring 2014 Page 13 Group Work
Class 2 page 13 Bohr Activity 5: continue 5B. Use the ata from part A to fill in the table for each value of n in the chart below. 5C. Using the ata from part B, plot the interactive energy of the electron vs. istance from the nucleus. Recall that you must raw a smooth curve. Energy vs. Distance for Charge Interaction n = = E = 1 2 3 4 Energy (Joules) 0-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9-1 -1.1-1.2 0 1 2 3 4 5 6 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 magnitue of the energy. What oes the graph tell you about the relationship between the orbit number n, the istance from the nucleus, an the magnitue of the energy? When the istance between the particles ecreases, the attractive interaction increases, the sign of the interactive energy is negative, the magnitue of the energy increases, an the overall energy ecreases. Spring 2014 Page 14 Group Work
Class 2 page 14 Bohr Activity 6: Energy Level Diagram 6A. Now construct a traitional energy level iagram for the Bohr moel. To simplify we will assume k=1 an = n 2. Use the numbers from the table you complete in Activity 5B. For each value of n, raw a line next to its corresponing energy on the scale below. Label each line with the value of n it represents. The line corresponing to n = 1 is rawn for you. Figure 1. Energy level iagram for k=1 E kq 1Q 2 0-0.10-0.20 Energy -0.30-0.40-0.50-0.60-0.70-0.80-0.90-1.00 n = 1 6B. What o you notice about the istance between the energy levels as n increases? 6C. As the value of n increases, the magnitue of the energy. 6D. As the value of n increases, the overall energy of the system. 6E. If the electron is completely remove from the system (n = ), what woul the overall energy of the system be? (Hint; Look at Activity 4F) Spring 2014 Page 15 Group Work
Class 2 page 15 Bohr Activity 6: continue. 6F. Compare your figure in 6A (Figure 1) to Figure 2 below. (Compare this to Figure F02-2-1 in the ebook). The lines you rew represent the allowe energy levels for the electron. The arrows on the iagram correspon to electronic transitions, which are covere in Activity 8. If you continue to a energy levels for greater values of n to figure 1, your iagram woul look more like Figure 2. Notice that if k=hcr H, the iagrams are the same. Each energy level on Figure 1 is multiplie by k, which we assume was 1 in orer to simplify our calculations. Each energy level on the iagram in Figure 2 is multiplie by the constant hcr H. Here h is Planck s constant, c is the spee of light, an R H is the Ryberg constant. Given the following information, what is the value of the constant hcr H? Be careful with canceling the units. h = 6.63 x 10-34 Js 0 c = 3.00 x 10 8 ms -1 1/16hcR H n = 4 R H = 1.096776 x 10 7 m -1 1/9 hcr H n = 3 n = 5 n = 6 k = hcr H = 1/4 hcr H n = 2 Energy Figure 2. Bohr moel energy level iagram for k=hcr H hcr H n = 1 Spring 2014 Page 16 Group Work
Class 2 page 16 Bohr Activity 6: continue. 6G. Use the previous activities to answer the following questions: i. Which electron has a greater magnitue 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 harer to remove from its orbit; n = 1 or n = 2? How o 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 magnitue. The lower the energy, the more negative the energy, an the more stable the state. 6H. Up to this point, we have assume that k=1 to simplify our calculations. Now, assume that k = hcr H, which is the number calculate in 6D. Also assume that = n 2, which is a goo approximation. Write out the equation that you woul 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 kq 1Q 2 simplify to obtain a general formula for E. This E is what the energy of the electron in a hyrogen atom is equal to, an can be use for to fin the energy for any value of n. Q 1 = 1 k = hcr H Q 2 = = n 2 E = This general equation is use to calculate the energy of an electron in any orbit in the H-atom. It is equation E02-2-6 in the e-book, with Z =. an YOU MUST HAVE YOUR TA INITIAL THIS PAGE BEFORE GOING ON. Spring 2014 Page 17 Group Work
Class 2 page 17 Bohr Activity 7: Energy of an Electron Note that aing 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 hyrogen atom. etc.... n = 1 n = 2 n = 3 n = 4 7A. The potential energy of an electron is relate to the interaction between the electron an 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 a 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, o 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 efinition of the energy of the electron. (Remember that the energy of the electron is negative. (= hcr H /n 2 ) 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 place into its orbit. Spring 2014 Page 18 Group Work
HOMEWORK : Bohr Moel Summary Name Chem 108 Section Use the following homework problems as a review to be sure you know the material in these activities. Activity 1; Energy vs. Distance a. charges attract. b. charges repel. c. As two opposite charges get closer together, the magnitue of the attractive energy.. As two like charges get closer together, the magnitue of the repulsive energy. Activity 2; Energy vs. Charge e. As charge increases, the magnitue of the attractive energy. f. As charge ecreases, the magnitue of the attractive energy. g. What happens to the repulsive energy between two like charges when the charges increase? Activity 3; Calculate E using a simplifie Coulomb s Law h. The equation for Coulomb s Law is. i. Energy is negative when the charges are. j. Energy is positive when the charges are. k. The sign of the energy is when the interaction is attractive. l. The sign of the energy is when the interaction is repulsive. m. The sign of the Energy of interaction provies information about. n. The absolute value of the Energy of interaction provies information about. Activity 4; Calculate Energy vs. Distance o. As the istance between particles gets larger, what happens to the magnitue of the energy of interaction? Spring 2014 Page 19 Group Work
Bohr Moel Summary HOMEWORK Name Chem 108 Section Activity 4; Calculate Energy vs. Distance p. When an interaction between oppositely charge particles is, the system is more stable, an the overall energy is. q. The strongest interaction has overall energy even though the magnitue of the energy is. Activity 5; Calculate Energy for Orbits r. As the value of n (the orbit number) increases, the magnitue of the energy s. As the value of n (the orbit number) increases, the overall interactive energy Activity 6; Simplifie Energy Level Diagrams t. What happens to the istance between the energy levels as n increases? u. What is the relationship between n (the orbit number) an (the istance between the electron an the nucleus)? v. Which electron is harer to remove from its orbit; n = 1 or n = 2? How o you know this? w. In which orbit oes an electron have higher overall energy: n = 1 or n = 5? x. What is the energy of the electron in a hyrogen atom equal to if k=hcr H an =n 2? Activity 7; Energy of an Electron m. The energy of the electron in the n th orbit is E hcr H. Describe in wors what this means; what is the energy of an electron? n 2 n. To completely remove an electron from the n = 1 orbit, o you put energy in or get energy out of the system? Note; You shoul also know how to construct an energy level iagram an how to raw an escribe electronic transitions between the levels. Spring 2014 Page 20 Group Work