Maxwell Relations: Difference between revisions
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'''Claimed by Ram Vempati (Fall 2024)''' | '''Claimed by Ram Vempati (Fall 2024)''' | ||
The Maxwell Relations are a set of partial derivative relations derived using [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] that enable the expression of physical quantities such as [https://en.wikipedia.org/wiki/Gibbs_free_energy Gibbs Free Energy] and [https://en.wikipedia.org/wiki/Enthalpy Enthalpy] as infinitesimal changes in pressure ( | The Maxwell Relations are a set of partial derivative relations derived using [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] that enable the expression of physical quantities such as [https://en.wikipedia.org/wiki/Gibbs_free_energy Gibbs Free Energy] and [https://en.wikipedia.org/wiki/Enthalpy Enthalpy] as infinitesimal changes in pressure ('''P'''), volume ('''V'''), temperature ('''T'''), and entropy ('''S'''). They are named after [[James Maxwell | James Maxwell]] and build upon the work done by [https://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann] in [https://en.wikipedia.org/wiki/Thermodynamics Thermodynamics] and [https://en.wikipedia.org/wiki/Statistical_mechanics Statistical Mechanics] | ||
== | ==Derivations== | ||
===Internal Energy=== | |||
The [https://en.wikipedia.org/wiki/First_law_of_thermodynamics first law of thermodynamics] states that <math>{\Delta U = \Delta Q - \Delta W}</math> where <math>{\Delta Q}</math> is the heat ''added to the system'' and <math>{\Delta W}</math> is the work done ''by the system''. We can re-express differentials Q and W with the [https://en.wikipedia.org/wiki/State_variable state variables] '''P''','''V''','''T''','''S''' using the substitutions <math>{dQ_{rev} = TdS}</math> (see [https://en.wikipedia.org/wiki/Clausius_theorem Clausius Theorem]) and <math>{dW = PdV}</math> (see [https://en.wikipedia.org/wiki/Work_(thermodynamics) Pressure-Volume Work]). We thus arrive at the thermodynamic definition for internal energy: <math>{dU = T dS − P dV}</math>. | |||
The Maxwell Relation for Internal Energy can be found by applying [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] to the system with the fundamental assumption that the variables can be expressed as [https://en.wikipedia.org/wiki/Exact_differential Exact Differentials] | |||
==Utility of Maxwell Relations== | |||
What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | ||
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== See also == | == See also == | ||
*[[Temperature & Entropy]] | |||
*[[Application of Statistics in Physics]] | |||
===Further reading=== | ===Further reading=== | ||
*[https://www.google.com/books/edition/Elements_of_Classical_Thermodynamics_For/GVhaSQ7eBQoC?hl=en Elements Of Classical Thermodynamics] | |||
===External | ===External Links=== | ||
==References== | ==References== | ||
Latest revision as of 12:03, 24 November 2024
Claimed by Ram Vempati (Fall 2024)
The Maxwell Relations are a set of partial derivative relations derived using Clairaut's Theorem that enable the expression of physical quantities such as Gibbs Free Energy and Enthalpy as infinitesimal changes in pressure (P), volume (V), temperature (T), and entropy (S). They are named after James Maxwell and build upon the work done by Ludwig Boltzmann in Thermodynamics and Statistical Mechanics
Derivations
Internal Energy
The first law of thermodynamics states that [math]\displaystyle{ {\Delta U = \Delta Q - \Delta W} }[/math] where [math]\displaystyle{ {\Delta Q} }[/math] is the heat added to the system and [math]\displaystyle{ {\Delta W} }[/math] is the work done by the system. We can re-express differentials Q and W with the state variables P,V,T,S using the substitutions [math]\displaystyle{ {dQ_{rev} = TdS} }[/math] (see Clausius Theorem) and [math]\displaystyle{ {dW = PdV} }[/math] (see Pressure-Volume Work). We thus arrive at the thermodynamic definition for internal energy: [math]\displaystyle{ {dU = T dS − P dV} }[/math].
The Maxwell Relation for Internal Energy can be found by applying Clairaut's Theorem to the system with the fundamental assumption that the variables can be expressed as Exact Differentials
Utility of Maxwell Relations
What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.
All Maxwell Relations
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See also
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References
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