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\sect{Absolute temperature and entropy}

In a general quasi static process we defined
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\[dQ\equiv dE+dW=\frac{\partial E}{\partial \beta}
d\beta+\left(\frac{\partial E}{\partial X}+y\right)dX\]
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We expressed ${E}$ and ${y}$ by the distribution function ${Z\left(\beta\right)}$

Using  differential equations technic, for an integration factor to
the non precise differential ${dQ}$,you learned in the course, which
is solely, a function of ${\beta}$ . Show that the integration
factor you get is
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\[T^{-1}\left(\beta\right)=\beta\]
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therefore It's possible to write ${dQ=TdS}$ where ${dS}$ is a
precise differential.

Find the function ${S}$ and show
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\[S=-\frac{\partial}{\partial
\left(\frac{1}{\beta}\right)}\left[-\frac{1}{\beta}\ln
Z\left(\beta\right)\right]\]
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By definition, ${S}$ is an entropy function, and ${T}$ is called in
an absolute temperature.



