We consider a thermodynamic system. If $$\Delta U$$ represents the increase in its internal energy and $$W$$ the work done by the system, which of the following statements is true ?

Solution :
An isothermal process is a constant temperature process. In this process, $$T = $$  constant or $$\Delta T = 0.$$
$$\eqalign{ & \therefore \Delta Q = \Delta U + \Delta W \cr & \Rightarrow \Delta Q = \Delta W\,\,\left( {\Delta U = 0} \right) \cr & \Delta U = n{C_V}\Delta T = 0 \cr} $$
An adiabatic process is defined as one with no heat transfer into or out of a system. Therefore, $$Q = 0.$$  From the first law of thermodynamics.
$$\eqalign{ & \Delta Q = \Delta U + \Delta W \cr & {\text{or}}\,\,\Delta U = - W\,\,\left[ {\Delta Q = 0} \right] \cr} $$