$(15$ marks$)$ You are provided $n$ cylinders containing different quantities of an ideal gas.

Each cylinder is fitted with a piston that is connected to a suitable reservoir whose pressure

is maintained constant. Let the $i^{th}$ cylinder contain $N_i$ moles of ideal gas, and let its piston

be connected to the reservoir at pressure $P_i.$ Let the volume of gas in the cylinder under the

above conditions be $V_{{?}_{?}}(i{)}^o.$ We plan to extract work from these cylinders by using a reversible

heat engine. Assume that we have figured out a way of transferring heat to$/$from one or more

of the cylinders. Let $C_p$ be the specific heat of the gas. We then perform the following steps:

Arrange the cylinders in the decreasing order of their initial temperatures. That is$,$ the

first cylinder has the highest temperature, and the $n^{th}$ cylinder has the lowest

temperature.

Connect the heat engine between first two cylinders and let it operate for a very long

time.

Connect the heat engine between the above two cylinders and the third cylinder, and let

it operate for a very long time.

Then, connect the heat engine between the above three cylinders, and the fourth cylinder,

and let it operate for a very long time. Continue this process till the $n^{th}$ cylinder.

Provide the expression for the final temperatures of the gas in the cylinders, and the total

work extracted from the engine.