Problem $3$: $(10$ pts$)$ Some theories for the origin of life on Earth suggest that early organisms obtained energy from the temperature gradient near ocean$-$floor hydrothermal vents. If the ambient water is at $4C$ and the hydrothermal vents are at $300C,$ consider the following.

$($a$)$ What is the highest efficiency possible when extracting work from these vents and the ambient water?

$($b$)$ If a hydrothermal vent leaks heat with a power of $2$ kW $,$ how much power can be extracted in the form of work?