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\section{Lois de laplace}

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	\begin{align*}
		\Delta U &= \delta W + \delta Q \\
			 &= \delta W &&\text{par adiabaticité} \\
			 &= -P \d V &&\text{par réversibilité} \\
		\iff C_V \d T &= -P \d V && \text{par caractère parfait du gaz} \\
		\iff \frac{nR}{\gamma-1} \d T &= -\left( \frac{nRT}{V} \right) \d V &&\text{par caractère parfait} \\
		\iff \frac{\d T}{T} \frac{nR}{\gamma-1} &= \frac{\d V}{V} (-nR) \\
		\iff \frac{\d T}{T} &= -(\gamma-1) \frac{\d V}{V} \\	
		\iff \ln T &= -\ln(V^{\gamma-1}) + \text{const}&&\text{par intégration} \\
		\iff TV^{\gamma-1} &= \text{const} \\
		\iff \frac{PV}{nR} V^{\gamma-1} &= \text{const} \\
		\iff PV^\gamma &= \text{const}&&\text{car $n, R$ constantes} \\
		\iff P \left( \frac{nRT}{P} \right)^{\gamma} &= \text{const} \\
		\iff P^{1-\gamma} T^\gamma &= \text{const} &&\text{car $n, R$ constantes} \\
	\end{align*}
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