# Heat exchange in adiabatic process

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? • The above equation does not say that the work done when a system is given heat at constant volume equals $nC_v\Delta T$. The relationship is not equal to the heat transferred. It is equal to the change in internal energy. Did you think that the only way the temperature of a gas can change is by adding or removing heat? – Chet Miller Oct 9 '19 at 3:07
• Adiabatic work is done in expense of thermal energy and vice versa. E.g air lifted by 100m adiabatically expands, does the expansion work and cools itself by 0.98 K. – Poutnik Oct 9 '19 at 3:26
• Please note that heat and temperature are completely different concepts. Do not confuse them. – Ezze Oct 9 '19 at 9:13

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.