Heat exchange in adiabatic process
The problem is that we were taught incorrectly in freshman physics.  They told us that $Q=nC\Delta T$, where C is called the heat capacity.  However, they neglected to tell us that this only applies if no work is done (i.e., the volume is essentially constant, as for a solid or liquid).  Otherwise, it gives the wrong answer for the heat Q.  

"Heat capacity" is a bit of a misnomer for C.  From what we then learn in thermodynamics, a better term would be the "internal energy capacity," because that is how Cv is more properly defined.  For an ideal gas,  $$nC_v=\frac{\Delta U}{\Delta T}$$ or $$\Delta U=nC_v\Delta T$$This has nothing to do with the heat added unless the volume is constant and no work is done.  In that case, $$Q=\Delta U=nC_v\Delta T$$So the freshman form of the equation is recovered only if no work is done.