Heat exchange in adiabatic process

Question

I know my teacher told me that

In adiabatic process there is no exchange for heat between system and surrounding.

Then he came to point. From first law of thermodynamics

$$∆U = q + W \tag{1}$$

Since in adiabatic process there is no heat exchange then,

$$∆U = W \tag{2}$$

And then he said as we know

$$∆U = nC_v∆T \tag{3}$$

So,

$$W = nC_v∆T \tag{4}$$

I don't get it. Above equation tells us that the work done when a system is given heat at constant volume equals to $nC_v∆T.$ But how can this be applied in adiabatic process, since there is no exchange of heat?

Answer 1

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.