Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow
In this paper, we formulate the first law of global thermodynamics for stationary states of the binary ideal gas mixture subjected to heat flow. We map the non-uniform system onto the uniform one and show that the internal energy <inline-formula><math xmlns="http://www.w3.org/1998/Math...
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| Format: | Article |
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MDPI AG
2023-10-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/11/1505 |
| _version_ | 1797459393794539520 |
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| author | Anna Maciołek Robert Hołyst Karol Makuch Konrad Giżyński Paweł J. Żuk |
| author_facet | Anna Maciołek Robert Hołyst Karol Makuch Konrad Giżyński Paweł J. Żuk |
| author_sort | Anna Maciołek |
| collection | DOAJ |
| description | In this paper, we formulate the first law of global thermodynamics for stationary states of the binary ideal gas mixture subjected to heat flow. We map the non-uniform system onto the uniform one and show that the internal energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><msup><mi>S</mi><mo>*</mo></msup><mo>,</mo><mi>V</mi><mo>,</mo><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><msub><mi>N</mi><mn>2</mn></msub><mo>,</mo><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> is the function of the following parameters of state: a non-equilibrium entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mo>*</mo></msup></semantics></math></inline-formula>, volume <i>V</i>, number of particles of the first component, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>1</mn></msub></semantics></math></inline-formula>, number of particles of the second component <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>2</mn></msub></semantics></math></inline-formula> and the renormalized degrees of freedom. The parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><msub><mi>N</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> satisfy the relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>/</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>+</mo><msub><mi>N</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>/</mo><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>2</mn></msub><mo>/</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>+</mo><msub><mi>N</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup><mo>/</mo><msub><mi>f</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>2</mn></msub></semantics></math></inline-formula> are the degrees of freedom for each component respectively). Thus, only 5 parameters of state describe the non-equilibrium state of the binary mixture in the heat flow. We calculate the non-equilibrium entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mo>*</mo></msup></semantics></math></inline-formula> and new thermodynamic parameters of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup></mrow></semantics></math></inline-formula> explicitly. The latter are responsible for heat generation due to the concentration gradients. The theory reduces to equilibrium thermodynamics, when the heat flux goes to zero. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties. |
| first_indexed | 2024-03-09T16:50:42Z |
| format | Article |
| id | doaj.art-3b40d2e7b0f94cf3837a97fcd5279f8b |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T16:50:42Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
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| spelling | doaj.art-3b40d2e7b0f94cf3837a97fcd5279f8b2023-11-24T14:40:57ZengMDPI AGEntropy1099-43002023-10-012511150510.3390/e25111505Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat FlowAnna Maciołek0Robert Hołyst1Karol Makuch2Konrad Giżyński3Paweł J. Żuk4Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, PolandInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, PolandIn this paper, we formulate the first law of global thermodynamics for stationary states of the binary ideal gas mixture subjected to heat flow. We map the non-uniform system onto the uniform one and show that the internal energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><msup><mi>S</mi><mo>*</mo></msup><mo>,</mo><mi>V</mi><mo>,</mo><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><msub><mi>N</mi><mn>2</mn></msub><mo>,</mo><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> is the function of the following parameters of state: a non-equilibrium entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mo>*</mo></msup></semantics></math></inline-formula>, volume <i>V</i>, number of particles of the first component, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>1</mn></msub></semantics></math></inline-formula>, number of particles of the second component <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>2</mn></msub></semantics></math></inline-formula> and the renormalized degrees of freedom. The parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><msub><mi>N</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> satisfy the relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>/</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>+</mo><msub><mi>N</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>/</mo><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>2</mn></msub><mo>/</mo><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>+</mo><msub><mi>N</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup><mo>/</mo><msub><mi>f</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>2</mn></msub></semantics></math></inline-formula> are the degrees of freedom for each component respectively). Thus, only 5 parameters of state describe the non-equilibrium state of the binary mixture in the heat flow. We calculate the non-equilibrium entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mo>*</mo></msup></semantics></math></inline-formula> and new thermodynamic parameters of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>f</mi><mn>1</mn><mo>*</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>*</mo></msubsup></mrow></semantics></math></inline-formula> explicitly. The latter are responsible for heat generation due to the concentration gradients. The theory reduces to equilibrium thermodynamics, when the heat flux goes to zero. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.https://www.mdpi.com/1099-4300/25/11/1505non-equilibrium thermodynamicsentropy and internal energymixtures and parameters of state |
| spellingShingle | Anna Maciołek Robert Hołyst Karol Makuch Konrad Giżyński Paweł J. Żuk Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow Entropy non-equilibrium thermodynamics entropy and internal energy mixtures and parameters of state |
| title | Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow |
| title_full | Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow |
| title_fullStr | Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow |
| title_full_unstemmed | Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow |
| title_short | Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow |
| title_sort | parameters of state in the global thermodynamics of binary ideal gas mixtures in a stationary heat flow |
| topic | non-equilibrium thermodynamics entropy and internal energy mixtures and parameters of state |
| url | https://www.mdpi.com/1099-4300/25/11/1505 |
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