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In his lecture (26:30-38:40), Shankar derives the adiabatic pressure-volume relationship $P_1V_1^\gamma = P_2V_2^\gamma$, where $\gamma = C_P / C_V$, from the First Law of Thermodynamics $\Delta U =Q - W$.
His first step in doing is is to make the substitution $\Delta U = n C_V \Delta T$ into the First Law. In adiabatic processes, volumes are not held constant, so why is using the specific heat at constant volume $C_V$ valid?
Glossary of Notation
$P$ - pressure
$V$ - volume
$T$ - temperature
$U$ - internal energy
$Q$ - heat added to system
$W$ - work done by system
$C_P$ - specific heat at constant pressure
$C_V$ - specific heat a constant volume
$n$ - number of moles