A Carnot engine operates between reservoirs at $500$ K and $300$ K$.$ Calculate its efficiency. If the engine absorbs $200$ J of heat per cycle, determine the work done.

Calculate the entropy change for $2$ moles of an ideal gas undergoing an isothermal reversible expansion from $V1=1\mathrm{.}0$ L to V$2=3\mathrm{.}0$ L at $298$ K $.$

Show that $(del\frac{S}{d}$elV${)}_{-}T=(del\frac{P}{d}$elT${)}_{-}V.$ Explain its physical significance.

A monatomic ideal gas initially at $300$ K and $2\mathrm{.}0$ atm undergoes reversible adiabatic expansion until its volume doubles. Determine the final temperature and pressure.

For a heat engine that operates on a Carnot cycle, sketch the T$-$S diagram. Calculate the total work done in one cycle if the entropy change between the hot $(373$ K $)$ and cold $(273$ K $)$ reservoirs is $0\mathrm{.}5\frac{J}{K}.$

Provide an example of an irreversible process. Calculate the total entropy change for the system and surroundings during an irreversible isothermal expansion.