Question

Internal energy in isochoric process is equal to work done in adiabatic process. Why does this happen?

Answer 1

An isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. An adiabatic process is one that occurs without transfer of heat or matter between a thermodynamic system and its surroundings. In an adiabatic process, energy is transferred only as work.

So if you put in energy or take out of an isochoric process the change will be in temperature and pressure of the system, which means the energy is transferred to the particles as their velocity or the kinetic energy of the system is increased. In an adiabatic process, there is no exchange of energy-matter or energy with the surroundings so all the work done is because of internal energy of the system.