$2\mathrm{.}0$ moles of an ideal gas are initially in a container of $45\mathrm{.}0d{m}^3$ at $298K.$ The container has a moveable piston covered with sand. Calculate the work performed for parts $($a$)$ through $($d$)$ below. All processes are isothermal.

a$)$ A scoop of sand is removed, and the gas expands to $65\mathrm{.}4d{m}^3.$ The final pressure of the gas is equal to the external pressure.

b$)$ The sand is added back onto the piston, and the gas compresses to $45\mathrm{.}0d{m}^3.$ The final pressure of the gas is equal to the external pressure.

c$)$ Sand is removed again, but this time, one grain at a time until the volume is $65\mathrm{.}4d{m}^3.$ The external pressure changes with each grain of sand removed.

d$)$ The sand is added back, but one grain at a time until the volume returns to $45\mathrm{.}0d{m}^3.$ The external pressure changes with each grain of sand added.