User:Tohline/PGE/FirstLawOfThermodynamics
First Law of Thermodynamics
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Following the detailed discussion of the laws of thermodynamics that can be found, for example in Chapter I of [C67] we know that, "for an infinitesimal quasi-statical change of state," the change <math>~dQ</math> in the total heat content <math>~Q</math> of a fluid element is given by, what we will label as,
Form A
of the First Law of Thermodyamics
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<math>~dQ</math> |
<math>~=</math> |
<math>~ d\epsilon + PdV \, , </math> |
[C67], Chapter II, Eq. (2)
where, <math>~\epsilon</math> is the specific internal energy, <math>~P</math> is the pressure, and <math>~V</math><math>~= 1/</math><math>~\rho</math> is the specific volume of the fluid element. Generally, the change in the total heat content can be rewritten in terms of the gas temperature, <math>~T</math>, and the specific entropy of the fluid, <math>~s</math>, via the expression,
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<math>~dQ</math> |
<math>~=</math> |
<math>~T ds \, .</math> |
[C67], Chapter II, Eq. (44)
If, in addition, it is understood that the specified changes are occurring over a certain interval of time <math>~dt</math>, then from this pair of expressions we derive what will henceforth be referred to as the,
Standard Form
of the First Law of Thermodyamics
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<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math> |
[C67], Chapter II, Eq. (2)
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