Students’ Ideas About the State-Function Property of Entropy*Warren M. Christensen, David E. Meltzer, Thomas A. Stroman

Iowa State University*Supported in part by NSF grants #DUE-9981140 and #PHY-0406724

P-V Diagram Cyclic Process

Calculus-Based Course (N = 341)

A B C D E

5% 23% 2% 67% 3%

Algebra-based Course (N = 232)

A B C D E

6% 19% 9% 62% 4%

The P-V diagram question was administered to all students in first-semester algebra-based and a calculus-based physics courses after all instruction was completed during the Spring 2005.

The cyclic process question was administered to second-semester calculus-based physics students in Spring 2005 after all instruction on thermodynamics was complete.

Cyclic Process Post-Instruction (N = 191)

a. Temperature b. Internal Energy c. Entropy d. Heat transfer

=0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0

89% 11% 74% 26% 54% 46% 40% 60%

This P-V diagram represents a system consisting of a fixed amount of ideal gas that undergoes three different processes in going from state A to state B:

Rank the change in entropy of the system for each process.NOTE: S1 represents the change in entropy of the system for Process #1, etc.

a. S3 < S2 < S1

b. S1 < S2 < S3

c. S1 = S2 < S3

d. S1 = S2 = S3

e. Not enough information

Solution: The entropy of a system is unique to its thermodynamic state; therefore, regardless of the process details, the change in entropy is the same for all three processes since they have the same initial and final states.

Student Interviews (N = 7): After receiving modified instruction on the state-function property of entropy, students were given the above problem.

Interview Data (N = 7)

S is the same for all processes 5Initially responded “S is path-dependent” but switched to “S is the same for all processes” 3 of 5

S is proportional to the area under the curve 2

A fundamental concept of thermodynamics is that a system in a particular state has a set of properties that is unique to that state. When a system changes from some initial state to some final state, the change of a given state function is the same regardless of how the system gets from the initial to the final state. Heat transfer, Q, is not a state function and its value depends on the process that the system undergoes. Student thinking regarding these quantities have been studied by Loverude, et al., [AJP, 2002] and Meltzer, [AJP, 2004] in the context of the first law of thermodynamics.

Is Q for Process #1 greater than, less than, or equal to that for Process #2?

Which would produce the largest change in the total energy of all the atoms in the system: Process #1, Process #2, or both processes produce the same change?

1999 2000 2001

Incorrect N = 186 N = 188 N = 279

Q1 = Q2 31% 43% 41%From Meltzer 2004

In 2001, 73% of students taking a second-semester calculus-based physics course (N = 279) determined correctly that the change in total energy would be the same for both processes.

Heat transfer is not a state function but ~40% of students give answers consistent with that idea.

65% of students were able to successfully answer this question. Since overgeneralization of the state function property has been seen in previous work in thermodynamics (see above), the high number of correct responses may not reflect a meaningful association between entropy and the state of a system. The most common incorrect answer is “b. S1 < S2 < S3”, which is consistent with the idea that greater area under the curve means greater entropy change and is probed further in one-on-one student interviews.

Response rates on this question are nearly identical to those reported by Meltzer on a similar question (above) for both internal energy always equal to zero [correct], 74% to 73%, and heat transfer always equal to zero [incorrect], 40% to 38%. However, students answer question (c) correctly [S always equal to zero] only 54% of the time, which is significantly different (p < 0.01) from the correct response rate (67%) in the question employing a P-V diagram.

Five out of the seven students asserted that the change in entropy would be greater for the process with the larger area under the curve. About half offered explicit reasoning using S ~ Q/T to justify that answer. After thinking about the problem further, three students changed to the correct answer, stating that they had just remembered it, and were confident they now had the correct answer.

Conclusions:• Student responses concerning the state-function property of entropy are significantly different when problems are posed using different representations.

• Students’ tendency to ascribe state-function properties to path-dependent quantities may mask their thinking about the state-function property of entropy.

Consider a heat engine that uses a fixed quantity of ideal gas. This gas undergoes a cyclic process which consists of a series of changes in pressure and temperature. The process is called “cyclic” because the gas system repeatedly returns to its original state (that is, same value of temperature, pressure, and volume) once per cycle.

Consider one complete cycle; the system begins in a certain state and returns to that same state, so the initial state and the final state are the same.

a) Is the change in temperature (T) of the gas at the completion of one complete cycle always equal to zero for any cyclic process or not always equal to zero for any cyclic process? Explain.

b) Is the change in internal energy (U) of the gas at the completion of one complete cycle always equal to zero for any cyclic process or not always equal to zero for any cyclic process? Explain.

c) Is the change in entropy (S) of the gas at the completion of one complete cycle always equal to zero for any cyclic process or not always equal to zero for any cyclic process? Explain.

d) Is the net heat transfer to the gas during one complete cycle always equal to zero for any cyclic process or not always equal to zero for any cyclic process? Explain.

T always = 0

U always = 0

S always = 0

QNET not always 0

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