Worksheet Boltzmann Distribution Applications

5/7/2021 Review Test Submission: Worksheet: Boltzmann Distribution ... https://uic.blackboard.com/webapps/assessment/review/review.jsp?attempt_id=_49252015_1&course_id=_193694_1&content_id=_8526424_1&outco… 1/7 Review Test Submission: Worksheet: Boltzmann Distribution Applications PHYS 260 Intro Thermal Physics (41588) 2021 Spring Weekly Content Week 9 begins Monday, March 8 Review Test Submission: Worksheet: Boltzmann Distribution Applications User Yacoub Awimrin Course PHYS 260 Intro Thermal Physics (41588) 2021 Spring Test Worksheet: Boltzmann Distribution Applications Started 3/8/21 8:42 PM Submitted 3/14/21 6:11 PM Status Completed Attempt Score 43 out of 55 points Time Elapsed 140 hours, 28 minutes Instructions Results Displayed All Answers, Submitted Answers, Correct Answers, Feedback, Incorrectly Answered Questions Units : Include units in your answer when instructed. Use SI units only in your answers. For example, use Pascals not atmospheres. Numbers : Use either the normal form, such as 132 or 132.5, or the exponential form, which represents 132 as 1.32E+2, represents 5 x 10 4 as 5E+4 or 5E4, or represents 52 x 10 -3 as 5.2E-2, unless instructed otherwise. Do not use the form 5 x 10 4 . If your answer is positive, do not place a "+" sign before your answer. Repeat Once : You can do worksheets twice before the due date. After submitting the ±rst time, click the OK on the bottom right of the page and you will be informed of your total score and which questions were answered incorrectly (during a one- time view, so you might want to take a screen-shot for future reference). Accept the score or redo the entire worksheet. Your highest score will be recorded. Answers and feedback to all problems will be available after the due date and can be accessed within My Grades. Question 1 Selected Answers: d. e. A harmonic oscillator with energy levels ε is in equilibrium with a reservoir at temperature T . Choose the correct statements below. The ground state has the highest probability of being occupied, independent of the temperature. If kT << ε , only the ground state is occupied. 5 out of 5 points

5/7/2021 Review Test Submission: Worksheet: Boltzmann Distribution ... https://uic.blackboard.com/webapps/assessment/review/review.jsp?attempt_id=_49252015_1&course_id=_193694_1&content_id=_8526424_1&outco… 2/7 Answers: a. b. c. d. e. Response Feedback: If kT >> ε , higher energy states are preferentially occupied over lower energy states. If kT >> ε , all states will be equally occupied. If kT = 2 ε , only the second excited state is occupied. The ground state has the highest probability of being occupied, independent of the temperature. If kT << ε , only the ground state is occupied. The ground state has the highest probability of being occupied. As the temperature is increased above kT = ε , the probability of occupying higher energy states increases. Question 2 Selected Answer: 0.932 Correct Answer: 0.932 ± 1% A molecule has states with the following energies: 0, 1 ε , 2 ε , 3 ε , and 4 ε , where ε = 0.9 x 10 -20 J. Calculate the probability of molecules in the ground state (with zero energy) for a collection of molecules in thermal equilibrium at T = 242 K. Provide your answer as a number in normal form to 3 decimal places (in the form X.XXX). Hint: note that this molecule has a ±nite number of states so you must take a ±nite sum, not the expressions discussed in the video for in±nite sums. Question 3 Selected Answer: 0.149 Correct Answer: 149 ± 5% Response Feedback: A molecule has states with the following energies: 0, 1 ε , 2 ε , 3 ε , and 4 ε , where ε = 0.5 x 10 -20 J. Calculate the average number of molecules in the ±rst excited state (1 ε ) for a collection of 1000 molecules in thermal equilibrium at T = 213 K. Note that the average number of molecules in a state is just the probability of the state times the number of molecules. Provide your answer as a number in normal form. An example calculation for T = 300 K and ε = 1 x 10 -20 J: 6 out of 6 points 0 out of 6 points

5/7/2021 Review Test Submission: Worksheet: Boltzmann Distribution ... https://uic.blackboard.com/webapps/assessment/review/review.jsp?attempt_id=_49252015_1&course_id=_193694_1&content_id=_8526424_1&outco… 4/7 Question 5 Selected Answers: d. e. Answers: a. b. c. d. e. The last slide of the video on applications of the Boltzmann distribution describes the behavior of vibrations and rotations of hydrogen (H 2 ) molecules in a gas at thermal equilibrium. Choose the correct statements below. Vibrational energy levels are separated by an energy ε ≈ 1000 k . Most hydrogen molecules in a gas at 2000 K are vibrating, but most do not vibrate at 500 K. The thermal energy is smaller than the energy di²erence between vibrational energy levels for T >> 1000 K. Hydrogen molecules in a gas will rotate if the temperature is in the range 100 K < T < 1000 K, but not outside that range. The energy separation between rotational energy levels is the same as the separation between vibrational energy levels. Vibrational energy levels are separated by an energy ε ≈ 1000 k . Most hydrogen molecules in a gas at 2000 K are vibrating, but most do not vibrate at 500 K. 6 out of 6 points

5/7/2021 Review Test Submission: Worksheet: Boltzmann Distribution ... https://uic.blackboard.com/webapps/assessment/review/review.jsp?attempt_id=_49252015_1&course_id=_193694_1&content_id=_8526424_1&outco… 5/7 Question 6 Selected Answer: 403.271636 Correct Answer: 403 ± 2% Calculate the most probable speed of oxygen (O 2 ) molecules in a gas at T = 313 K, see the handout continuous_levels.pdf. O 2 atomic mass is 32 amu, where the atomic mass unit (amu) is 1.66 x 10 -27 kg. Provide your answer in units of meters per second, but do not include the units in your answer, just the number in normal form. Question 7 Selected Answer: 408.30137 Correct Answer: 408 ± 2% Calculate the average speed of oxygen (O 2 ) molecules in a gas at T = 252 K, see the handout continuous_levels.pdf. O 2 atomic mass is 32 amu, where the atomic mass unit (amu) is 1.66 x 10 -27 kg. Provide your answer in units of meters per second, but do not include the units in your answer, just the number in normal form. Question 8 The moon's atmosphere is much less dense than the earth's. One contributing factor is the reduced level of gravity on the moon, so that the gravitational attraction of gas molecules to it is smaller. Let's pretend that the moon has an oxygen (O 2 ) atmosphere at T = 300 K. Calculate the escape speed for an oxygen molecule from the moon. This is really a mechanics problem, so let me remind you how to do it. We'll address the thermal physics of this situation in a later problem. A molecule can escape from the moon when its kinetic energy ½ mv 2 is equal to or greater than its gravitational potential energy near the moon, U pot = - m molecule g moon r moon . The “escape speed” is the minimum speed that allows it to escape. Equating these two energies, but excluding the "–" sign, allows you to calculate the escape speed because it provides the minimum kinetic energy required to escape the gravity of the moon. The gravitational acceleration on the moon g moon = 1.6 m s -2 and the radius of the moon, r moon = 1700 kilometers, the atomic mass of O 2 is 32 amu, and the atomic 6 out of 6 points 6 out of 6 points 5 out of 5 points

5/7/2021 Review Test Submission: Worksheet: Boltzmann Distribution ... https://uic.blackboard.com/webapps/assessment/review/review.jsp?attempt_id=_49252015_1&course_id=_193694_1&content_id=_8526424_1&outco… 7/7 Friday, May 7, 2021 3:36:27 PM CDT Selected Answer: 2.14E-13 Correct Answer: 2.3E-13 ± 50% Response Feedback: where “x” represents the velocity and you will need to consider the integral in the beginning of this problem carefully in order to replace the constant terms A and B with numerical values. The value of this integral is the probability that the speed is equal to or greater than the "fake" escape speed of 2200 m/s, but the answer is not far o² from the one you would get if you used the correct escape speed. Enter your numerical answer in exponential form as described in the worksheet instructions. ← OK