Editing PGE/FirstLawOfThermodynamics (section)

==Standard Presentation==
Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] 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 the,
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<span id="FundamentalLaw"><font color="#770000">'''Fundamental Law of Thermodynamics'''</font></span>

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  <td align="right">
<math>dQ</math>
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<math>=</math>
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<math>
d\epsilon + PdV \, ,
</math>
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</tr>
</table>
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (2)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, Eq. (1.2)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;4.1, Eq. (4.1)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;1.6.5, Eq. (1.124)
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where, {{ Template:Math/VAR_SpecificInternalEnergy01 }} is the specific internal energy, {{ Template:Math/VAR_Pressure01 }} is the pressure, and {{ Template:Math/VAR_SpecificVolume01 }} <math>= 1/</math>{{ Template:Math/VAR_Density01 }} 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, {{ Template:Math/VAR_Temperature01 }}, and the specific entropy of the fluid, {{ Template:Math/VAR_SpecificEntropy01 }}, via the expression,
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<math>~dQ</math>
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<math>~=</math>
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<math>~T ds \, .</math>
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[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter I, Eq. (76) &amp; Chapter II, Eq. (44)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.4, p. 16<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)
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<span id="VariableDimensions"><table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
<div align="center">'''Variable Dimensions'''</div>
Each of the terms in these two expressions has units of specific energy, that is, energy per unit mass.  Specifically, these are the units for the two variables, <math>Q</math> and {{ Template:Math/VAR_SpecificInternalEnergy01 }}, while the product of {{ Template:Math/VAR_Pressure01 }} (energy per unit volume) and {{ Template:Math/VAR_SpecificVolume01 }} (volume per unit mass) gives specific energy.  It should be clear as well that {{ Template:Math/VAR_SpecificEntropy01 }} has units of specific energy per Kelvin; given that {{ Template:Math/VAR_SpecificEntropy01 }} is usually referred to in the literature as "specific entropy," we conclude that entropy, itself, has units of energy per Kelvin.
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If, in addition, it is understood that the specified changes are occurring over an interval of time d{{ Template:Math/VAR_Time01 }}, then from this pair of expressions we derive what will henceforth be referred to as the,

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<span id="Standard Form"><font color="#770000">'''Standard Form'''</font></span><br />
of the First Law of Thermodyamics

{{ Template:Math/EQ_FirstLaw01 }}

[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (64)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Chapter 4, Eq. (4.27)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.3.3, Eq. (7.162)
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If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then <math>ds/dt = 0</math> and the changes are said to have been made ''adiabatically.''  For an adiabatically evolving system, therefore, the ''First Law'' assumes what henceforth will be referred to as the,

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<span id="Standard Form"><font color="#770000">'''Adiabatic Form'''</font></span><br />
of the First Law of Thermodyamics

{{ Template:Math/EQ_FirstLaw02 }}

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (13)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (70)
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Clearly this form of the ''First Law'' also may be viewed as a statement of ''specific entropy conservation.''