Quirks of Stirling's Approximation
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| Title: | Quirks of Stirling's Approximation |
|---|---|
| Language: | English |
| Authors: | Macrae, Roderick M., Allgeier, Benjamin M. |
| Source: | Journal of Chemical Education. Jun 2013 90(6):731-734. |
| Availability: | Division of Chemical Education, Inc and ACS Publications Division of the American Chemical Society. 1155 Sixteenth Street NW, Washington, DC 20036. Tel: 800-227-5558; Tel: 202-872-4600; e-mail: eic@jce.acs.org; Web site: http://pubs.acs.org/jchemeduc |
| Peer Reviewed: | Y |
| Page Count: | 4 |
| Publication Date: | 2013 |
| Document Type: | Journal Articles Reports - Descriptive |
| Education Level: | Higher Education Postsecondary Education |
| Descriptors: | Science Instruction, College Science, Undergraduate Study, Physical Sciences, Chemistry, Scientific Concepts, Misconceptions, Equations (Mathematics), Thermodynamics |
| DOI: | 10.1021/ed300560w |
| ISSN: | 0021-9584 |
| Abstract: | Stirling's approximation to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy model" example, the two-state system of "N" classical spins, where it is shown that two different physical situations lead to the same computed value of the entropy. Retention of additional terms in the approximation of the factorial is required to yield an accurate expression for the statistical weight of the most probable configuration in such model systems, but generates only a little extra accuracy in entropy calculations, and then only in the limit of very small numbers of particles. Additionally, inclusion of these terms makes the entropy nonextensive. We show here that, in the standard derivation of the entropy of the microcanonical ensemble, it is the freedom to allow the ensemble size to be infinite that makes the Boltzmann entropy expression S = k[subscript B] ln"W" exact, a fact that is not widely understood. |
| Abstractor: | As Provided |
| Number of References: | 11 |
| Entry Date: | 2014 |
| Accession Number: | EJ1017061 |
| Database: | ERIC |
| FullText | Text: Availability: 0 |
|---|---|
| Header | DbId: eric DbLabel: ERIC An: EJ1017061 AccessLevel: 3 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Quirks of Stirling's Approximation – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Macrae%2C+Roderick+M%2E%22">Macrae, Roderick M.</searchLink><br /><searchLink fieldCode="AR" term="%22Allgeier%2C+Benjamin+M%2E%22">Allgeier, Benjamin M.</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Journal+of+Chemical+Education%22"><i>Journal of Chemical Education</i></searchLink>. Jun 2013 90(6):731-734. – Name: Avail Label: Availability Group: Avail Data: Division of Chemical Education, Inc and ACS Publications Division of the American Chemical Society. 1155 Sixteenth Street NW, Washington, DC 20036. Tel: 800-227-5558; Tel: 202-872-4600; e-mail: eic@jce.acs.org; Web site: http://pubs.acs.org/jchemeduc – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: Pages Label: Page Count Group: Src Data: 4 – Name: DatePubCY Label: Publication Date Group: Date Data: 2013 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Descriptive – Name: Audience Label: Education Level Group: Audnce Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Science+Instruction%22">Science Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22College+Science%22">College Science</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Study%22">Undergraduate Study</searchLink><br /><searchLink fieldCode="DE" term="%22Physical+Sciences%22">Physical Sciences</searchLink><br /><searchLink fieldCode="DE" term="%22Chemistry%22">Chemistry</searchLink><br /><searchLink fieldCode="DE" term="%22Scientific+Concepts%22">Scientific Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Misconceptions%22">Misconceptions</searchLink><br /><searchLink fieldCode="DE" term="%22Equations+%28Mathematics%29%22">Equations (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Thermodynamics%22">Thermodynamics</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1021/ed300560w – Name: ISSN Label: ISSN Group: ISSN Data: 0021-9584 – Name: Abstract Label: Abstract Group: Ab Data: Stirling's approximation to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy model" example, the two-state system of "N" classical spins, where it is shown that two different physical situations lead to the same computed value of the entropy. Retention of additional terms in the approximation of the factorial is required to yield an accurate expression for the statistical weight of the most probable configuration in such model systems, but generates only a little extra accuracy in entropy calculations, and then only in the limit of very small numbers of particles. Additionally, inclusion of these terms makes the entropy nonextensive. We show here that, in the standard derivation of the entropy of the microcanonical ensemble, it is the freedom to allow the ensemble size to be infinite that makes the Boltzmann entropy expression S = k[subscript B] ln"W" exact, a fact that is not widely understood. – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: Ref Label: Number of References Group: RefInfo Data: 11 – Name: DateEntry Label: Entry Date Group: Date Data: 2014 – Name: AN Label: Accession Number Group: ID Data: EJ1017061 |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1017061 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1021/ed300560w Languages: – Text: English PhysicalDescription: Pagination: PageCount: 4 StartPage: 731 Subjects: – SubjectFull: Science Instruction Type: general – SubjectFull: College Science Type: general – SubjectFull: Undergraduate Study Type: general – SubjectFull: Physical Sciences Type: general – SubjectFull: Chemistry Type: general – SubjectFull: Scientific Concepts Type: general – SubjectFull: Misconceptions Type: general – SubjectFull: Equations (Mathematics) Type: general – SubjectFull: Thermodynamics Type: general Titles: – TitleFull: Quirks of Stirling's Approximation Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Macrae, Roderick M. – PersonEntity: Name: NameFull: Allgeier, Benjamin M. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Type: published Y: 2013 Identifiers: – Type: issn-print Value: 0021-9584 Numbering: – Type: volume Value: 90 – Type: issue Value: 6 Titles: – TitleFull: Journal of Chemical Education Type: main |
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