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Change of Entropy During Ideal Gas Processes

Experiments show that the integral  
\[\int_C \frac{1}{T} dU + \frac{p}{T} dV\]
  is independent of the path.
Since the integral is indeoendent of the path we can define a scalar function  
\[S = \int_C \frac{1}{T} dU + \frac{p}{T} dV \]

This scalar is called the entropy and is a measure of how disordered the gas is and the integral gives the change in  
\[S\]
  along  
\[C\]
.
\[U=U(p,V) \rightarrow dU = \frac{\partial U}{\partial p} dp + \frac{\partial U}{\partial V} dV \]

Hence
\[\begin{equation} \begin{aligned} S &= \int_C \frac{1}{T} dU + \frac{p}{T} dV = \\ &= \int_C (\frac{1}{T} (\frac{\partial U}{\partial p} dp + \frac{\partial U}{\partial V} dV ))+ \frac{p}{T} dV \\ &= \int_C \frac{1}{T} \frac{\partial U}{\partial p} dp + \frac{1}{T} (\frac{\partial U}{\partial V} + p ) dV \end{aligned} \end{equation}\]

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