Why is $C_V$ used in this derivation? 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
 A: $C_v$ is defined for a gas by$$Q=nC_v\Delta T$$
in which $Q$ is the heat inflow needed to raise the temperature of the gas by $\Delta T$ at constant volume.
At constant volume, no work is done, so the First Law collapses to
$$\Delta U = Q.$$
Therfore$$\Delta U = nC_v\Delta T.$$
But for an ideal gas, $U$ is proportional to $n$ and $T$, whatever sort of process the gas is undergoing. There is therefore a process-independent proportionality constant. But we know from the constant volume process that this constant is equal to $C_v$ ! We name the constant after just one of the roles that it plays.
[This is a perennial cause of confusion to students learning thermodynamics. I find the following comparable case instructive…
The fundamental role of the space-time constant, $c,$ is as a scale factor between times and displacements, so, for example we can write the invariant interval as$$(c\Delta \tau)^2=(c\Delta t)^2-[(\Delta x)^2+(\Delta y)^2+(\Delta z)^2]$$Yet we call $c$ "the speed of light", after one of the roles that it plays.] 
A: He started with the first law
$\Delta U = \Delta Q - P \Delta V$
and then noted that $\Delta Q = 0$ for the adiabatic process, therefore we have
$0 = \Delta U + P \Delta V$
Now, for an ideal gas, we know that $\Delta U = nC_v \Delta T$. Therefore 
$0 = nC_v \Delta T + P \Delta V$
and this was the basis of the rest of his derivation.
