## 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{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}