d U = T d S - p d V
Solve for S
\left\{\begin{matrix}S=\frac{Vp+U}{T}\text{, }&T\neq 0\\S\in \mathrm{R}\text{, }&d=0\text{ or }\left(U=-Vp\text{ and }T=0\right)\end{matrix}\right.
Solve for T
\left\{\begin{matrix}T=\frac{Vp+U}{S}\text{, }&S\neq 0\\T\in \mathrm{R}\text{, }&\left(U=-Vp\text{ and }S=0\right)\text{ or }d=0\end{matrix}\right.
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TdS-pdV=dU
Swap sides so that all variable terms are on the left hand side.
TdS=dU+pdV
Add pdV to both sides.
TdS=Vdp+Ud
The equation is in standard form.
\frac{TdS}{Td}=\frac{d\left(Vp+U\right)}{Td}
Divide both sides by Td.
S=\frac{d\left(Vp+U\right)}{Td}
Dividing by Td undoes the multiplication by Td.
S=\frac{Vp+U}{T}
Divide d\left(U+pV\right) by Td.
TdS-pdV=dU
Swap sides so that all variable terms are on the left hand side.
TdS=dU+pdV
Add pdV to both sides.
SdT=Vdp+Ud
The equation is in standard form.
\frac{SdT}{Sd}=\frac{d\left(Vp+U\right)}{Sd}
Divide both sides by dS.
T=\frac{d\left(Vp+U\right)}{Sd}
Dividing by dS undoes the multiplication by dS.
T=\frac{Vp+U}{S}
Divide d\left(U+pV\right) by dS.
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