I was reading the definition of heat capacity and it says that
$$\Delta Q = C \ \Delta T$$
(introduction to statistical physics – K. Huang)
So my question is that, if we consider an isothermal process, becuase temperature remains the same, $\Delta Q$ would be zero. And $\Delta U= \Delta W$ by the first law, but thats wrong too because of what I have read on the internet. So, what am I doing wrong? Why do you have in a isothermal process $W=Q$ and not $U=W$?
Best Answer
Suppose you take an ideal gas as your system. Then according to the equipartition theorem its total internal energy would be $\frac{1}{2}k_B T$ per degree of freedom per molecule. So if $f$ is the number of degrees of freedom then the total internal energy of your $N$ molecule ideal gas system would be $$\frac{f}{2}NK_BT$$ or $$\frac{f}{2}nRT$$
So as you can see the total internal energy only depends on the temperature. And moreover if $\Delta T$ is the change in temperature in a process then the corresponding change in total internal energy would be $$\Delta U = \frac{f}{2}nR\Delta T$$
As in isothermal process temperature remains constant, $\Delta T= 0 $.
So $$\Delta U =0$$
And the first of law of thermodynamics becomes $$W= -Q$$
(PS: Also, I want to point out a flaw in your question: Even when $T$ remains same, $\Delta Q$ need not be zero. Think of what happens in a phase change situation i.e latent heat.)