Adsorption of an Ideal Gas on a Small Spherical Adsorbent
The ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be modeled as ideal for low densities. The adsorption energy,...
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| Format: | Article |
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MDPI AG
2021-02-01
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| Series: | Nanomaterials |
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| Online Access: | https://www.mdpi.com/2079-4991/11/2/431 |
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| author | Bjørn A. Strøm Dick Bedeaux Sondre K. Schnell |
| author_facet | Bjørn A. Strøm Dick Bedeaux Sondre K. Schnell |
| author_sort | Bjørn A. Strøm |
| collection | DOAJ |
| description | The ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be modeled as ideal for low densities. The adsorption energy, <inline-formula><math display="inline"><semantics><msup><mi>U</mi><mi>s</mi></msup></semantics></math></inline-formula>, plays an important role in the analysis. For small adsorbents this energy depends on the curvature of the adsorbent. We model the adsorbent as a sphere with surface area <inline-formula><math display="inline"><semantics><mrow><mo>Ω</mo><mo>=</mo><mn>4</mn><mi>π</mi><msup><mi>R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, where <i>R</i> is the radius of the sphere. We calculate the partition function for a grand canonical ensemble of two-dimensional adsorbed phases. When connected with the nanothermodynamic framework this gives us the relevant thermodynamic variables for the adsorbed phase controlled by the temperature <i>T</i>, surface area <inline-formula><math display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, and chemical potential <inline-formula><math display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>. The dependence of intensive variables on size may then be systematically investigated starting from the simplest model, namely the ideal adsorbed phase. This dependence is a characteristic feature of small systems which is naturally expressed by the subdivision potential of nanothermodynamics. For surface problems, the nanothermodynamic approach is different, but equivalent to Gibbs’ surface thermodynamics. It is however a general approach to the thermodynamics of small systems, and may therefore be applied to systems that do not have well defined surfaces. It is therefore desirable and useful to improve our basic understanding of nanothermodynamics. |
| first_indexed | 2024-03-09T05:03:28Z |
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| id | doaj.art-9fe2cb31bcda44c397d974656f868b98 |
| institution | Directory Open Access Journal |
| issn | 2079-4991 |
| language | English |
| last_indexed | 2024-03-09T05:03:28Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Nanomaterials |
| spelling | doaj.art-9fe2cb31bcda44c397d974656f868b982023-12-03T12:57:27ZengMDPI AGNanomaterials2079-49912021-02-0111243110.3390/nano11020431Adsorption of an Ideal Gas on a Small Spherical AdsorbentBjørn A. Strøm0Dick Bedeaux1Sondre K. Schnell2Department of Materials Science and Engineering, Faculty of Natural Sciences, Norwegian University of Science and Technology, NTNU, NO-7491 Trondheim, NorwayPorelab, Department of Chemistry, Norwegian University of Science and Technology, NTNU, NO-7491 Trondheim, NorwayDepartment of Materials Science and Engineering, Faculty of Natural Sciences, Norwegian University of Science and Technology, NTNU, NO-7491 Trondheim, NorwayThe ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be modeled as ideal for low densities. The adsorption energy, <inline-formula><math display="inline"><semantics><msup><mi>U</mi><mi>s</mi></msup></semantics></math></inline-formula>, plays an important role in the analysis. For small adsorbents this energy depends on the curvature of the adsorbent. We model the adsorbent as a sphere with surface area <inline-formula><math display="inline"><semantics><mrow><mo>Ω</mo><mo>=</mo><mn>4</mn><mi>π</mi><msup><mi>R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, where <i>R</i> is the radius of the sphere. We calculate the partition function for a grand canonical ensemble of two-dimensional adsorbed phases. When connected with the nanothermodynamic framework this gives us the relevant thermodynamic variables for the adsorbed phase controlled by the temperature <i>T</i>, surface area <inline-formula><math display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, and chemical potential <inline-formula><math display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>. The dependence of intensive variables on size may then be systematically investigated starting from the simplest model, namely the ideal adsorbed phase. This dependence is a characteristic feature of small systems which is naturally expressed by the subdivision potential of nanothermodynamics. For surface problems, the nanothermodynamic approach is different, but equivalent to Gibbs’ surface thermodynamics. It is however a general approach to the thermodynamics of small systems, and may therefore be applied to systems that do not have well defined surfaces. It is therefore desirable and useful to improve our basic understanding of nanothermodynamics.https://www.mdpi.com/2079-4991/11/2/431adsorptionnanothermodynamicssmall-systemsize-dependentthermodynamicsstatistical mechanics |
| spellingShingle | Bjørn A. Strøm Dick Bedeaux Sondre K. Schnell Adsorption of an Ideal Gas on a Small Spherical Adsorbent Nanomaterials adsorption nanothermodynamics small-system size-dependent thermodynamics statistical mechanics |
| title | Adsorption of an Ideal Gas on a Small Spherical Adsorbent |
| title_full | Adsorption of an Ideal Gas on a Small Spherical Adsorbent |
| title_fullStr | Adsorption of an Ideal Gas on a Small Spherical Adsorbent |
| title_full_unstemmed | Adsorption of an Ideal Gas on a Small Spherical Adsorbent |
| title_short | Adsorption of an Ideal Gas on a Small Spherical Adsorbent |
| title_sort | adsorption of an ideal gas on a small spherical adsorbent |
| topic | adsorption nanothermodynamics small-system size-dependent thermodynamics statistical mechanics |
| url | https://www.mdpi.com/2079-4991/11/2/431 |
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