I am extremely confused on the concept of a $pV$ diagram.
I understand that Pressure times (change in) Volume equals the Work done by a piston; however, I am confused on how volume and pressure can increase at the same time. According to Boyle's Law, PV=k under a constant temperature. If this is so, how would volume and pressure increase at the same time in a closed system? Is it because heat (Q) is being added or taken away?
Thermodynamics is inherently multivariable. Everything depends on multiple parameters, but several different parameters can all be adjusted independently.
If I take a fixed-size container and increase $T$, then $p$ increases. If I make the container adjustable and decrease $V$ while maintaining $T$, the pressure increases. Those two very different processes could have the same end result on the pressure. Just because a process is graphed on a $p-V$ plane doesn't mean other variables can't change--we simply can't draw graphs on axes with enough dimensions to show everything simultaneously. The question you should always ask yourself with every thermodynamic process is "which variables are controlled and which are allowed to change in reaction?"
Start with $n$ moles of an ideal gas in a cylinder fitted with a piston. Initially we have $P_1$, $T_1$ and $V_1$.
Increase pressure by heating at constant volume: now $P_2>P_1$ but volume stays the same.
Next start heating at constant pressure, i.e., let the piston expand as heat is added while opposing the expansion with constant pressure $P_2$. In the final state both $P$ and $V$ will be higher.
