4. Carnot's model for an engine, which turns heat into useful work, consists of a hot reservoir (temperature Thot) and a cold reservoir (Teold). A certain amount of heat is extracted from the hot reservoir: some of this heat is converted by the engine into useful work and some is rejected (dumped) into the cold reservoir. At first sight, it seems wasteful to dump heat - why not convert it all to work? However, the Second Law tells us that to make the process spontaneous (i.e. to make it work at all) we must dump a certain amount of heat into the cold reservoir. Removing heat from the hot reservoir decreases its entropy but dumping heat into the cold reservoir increases its entropy. To make the overall process spontaneous, the entropy of the Universe (which in this case is composed of just the two reservoirs, which we assume to be perfectly insulated) must increase. Suppose that the hot reservoir is at 100 C and the cold at 25 C; suppose also that the reservoirs are infinitely large. (4.1) If 100 J of heat is extracted from the hot reservoir, what is the entropy change of this reservoir? (4.2) How much heat would need to be dumped into the cold reservoir to compensate exactly for the entropy lost when the 100 J is extracted from the hot reservoir? Hence, calculate the maximum amount of work that can be done by the engine. (4.3) Would the engine run if we tried to extract (i) more, or (ii) less work than the amount calculated in (4.2)? Give your reasons. (4.4) How could the engine be altered so that more of the 100 J of heat is converted to work? (4.5) If the amount of heat leaving the hot reservoir is qhot, the amount dumped into the cold reservoir is qcold and the work extracted is w, show that the maximum possible efficiency, n, is given by Thot - Tcold = where n is defined as n = Phot W Thot