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10.2: The Joule Experiment
https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Heat_and_Thermodynamics_(Tatum)/10%3A_The_Joule_and_Joule-Thomson_Experiments/10.02%3A_The_Joule_Experiment
\[\begin{align} d S &=\frac{1}{T}[d U+P d V] \\[4pt] &=\dfrac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T} d V+\left(\frac{\partial U}{\partial T}\right)_{V} d T+P d V\right].\\[4pt] &=\f... $\begin{align} d S &=\frac{1}{T}[d U+P d V] \\[4pt] &=\dfrac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T} d V+\left(\frac{\partial U}{\partial T}\right)_{V} d T+P d V\right].\\[4pt] &=\frac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T}+P\right] d V+\frac{1}{T}\left(\frac{\partial U}{\partial T}\right)_{V} d T. \end{align}$

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