$($a$)$ Define the terms

i$.$ adiabatic process

ii$.$ isothermal process

and state the relationship between pressure and $P$ and volume $V$ for an ideal gas for each of these processes.

$($b$)$ In addition to the above there are two more characteristic thermodynamic processes that can take place:

i$.$ isochor $-$ a thermodynamic change at constant volume

ii$.$ isobar $-$ a thermodynamic change at constant pressure

In the following we will consider a thermodynamic cycle composed out of the following four paths where A$,$ B$,$ C and D label thermodynamic states.

Initially the system which is an ideal gas is at temperature $T_{AB}$ in state $A$ in thermal contact with a heat bath at the same temperature.

$AB$ : The first change is an isothermal expansion to state $B.$

B $$ C : At point B the system is brought in thermal contact with a second colder heat bath at temperature $T_{CD}.$ Consequently it is cooling along an isochor to a new thermodynamic state $C.$

C$$ D: This is followed by an isothermal compression until the volume of state $A$ is reached.

$DA$: The final step is bringing the sytem now again in thermal contact with the hotter heat bath at temperature $T_{AB}$ and allowing it to warm up in along an isochor to reach its initial state $A$

Draw this cycle in a $PV$ diagram similar to that of the Carnot cycle