Student understanding of the Boltzmann factor

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students’ abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students’ appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.

Figure 1

Sample system for the Boltzmann factor instructional sequence. An isolated container of an ideal gas is separated into a small system ($C$) and a large reservoir ($R$). The label “$C$” is used to avoid confusion with entropy.

Figure 2

Probability ratios question (PRQ) given as an ungraded survey before tutorial instruction.

Figure 3

PRQ pretutorial results. (a) School 1, after lecture instruction on the Boltzmann factor over seven years ($N=50$); (b) school 2, before any instruction on the Boltzmann factor in one year ($N=32$). The green diagonal stripes indicate the students who used the Boltzmann factor or stated that the energy of the ground state was irrelevant (in part B) to obtain their chosen answers. Students in the “Other” column often provided no explicit answer or stated that there was not enough information to determine the answer. Only 24 students from school 1 and four students from school 2 used the Boltzmann factor on both parts.

Figure 4

PRQ Analog developed by instructor at school 2. Administered on a course exam once at school 1 and at school 2.

Figure 5

Post-tutorial results from PRQ and PRQ Analog, administered during course examinations. The green diagonal stripes indicate the students who used the Boltzmann factor or stated that the energy of the ground state was irrelevant (in part B) to obtain their chosen answer(s). For the PRQ Analog, “Equal to” corresponds to choice II, “Greater than” to choice I, “Less than” to choice III, and “Other” to choice IV. Data are shown for students who completed the PRQ pretest, participated in the Boltzmann Factor tutorial, and completed either the PRQ or the PRQ Analog exam question.