Isothermal paradox(kind of)
In freshman physics, we were taught that $q=mC\Delta T$, but in all the cases we considered, no work was being done.  When we got to thermodynamics, we learned that q represents energy in transit, and is a function of process path.  We also learned that, when work is being done, q is not equal to $mC\Delta T$.  Instead, we learned about the internal energy of a system U, which is expressed as $$\Delta U=q-w$$  When no work is being done, $$\Delta U=q=mC\Delta T$$All this misunderstanding is resolved and made consistent in thermodynamics when we change the definition of heat capacity slightly, defining it strictly in terms of U, rather than in terms of q:$$C_v=\left(\frac{\partial U}{\partial T}\right)_V$$Then, whether or not work is being done, we get the correct answer for the heat transferred and the change in internal energy.  The advantage of this definition is that C is properly recognized as a physical property of the material being processed rather than a function of the process path.