Problem Three:

"A gas is confined to a vertical cylinder by a piston of mass 2 kg and radius 1 cm. When 5 J of heat are added, the piston rises by 2.4 cm. Find: (a) the work done by the gas; (b) the change in its internal energy. Atmospheric pressure is Pa."

(a) P=Force/Area + one atmosphere N/m²

J

(b) J

*From Chapter 21, "Entropy and the Second Law of Thermodynamics:"

1. A heat engine produces work because of, in the most general sense, the heat difference (potential) between a hot and a cold reservoir. In natural, real processes, heat always flows from hot to cold, never the other way around (without the additional input of work). The second law of thermodynamics accounts for this; its first formulations pertained specifically to the work derived from heat engines. The work potential is defined as , where is the quantity of heat leaving the hot reservoir, in a given cycle, and is the amount of non-working heat delivered (and lost) to the cold reservoir.

2. The efficiency of a heat engine is defined as work output over heat input, or . In theory, all of the heat input could be turned into work if were 0, i.e., if there was no heat delivered to the cold reservoir. In fact, this is an impossibility. As Carnot, Clausius, and others have shown through relentless experimentation, a heat engine's efficiency is always less than one; that is, for every work transaction there is a loss.

3. That efficiency is always less than one is the second law in a nut shell; the loss represented by the less-than-one ratio is called entropy. The change in entropy (S), a state variable (independent of thermodynamic path), is defined as the ratio of heat transfer to temperature (temperature difference between reservoirs, states, etc.): , where is a reversible (discrete and definable) heat transfer.

In this sense, the entropy law says , or just for any real process. A "real" process is taken to mean one in which there is no thermal isolation from surroundings; hence, there is a heat transfer. "Adiabatic" is used to describe a process that involves no heat transfer; while a diesel engine power stroke is one example of a process said to be adiabatic, in fact it is only approximately adiabatic--there is in fact heat transfer.

Problem Four:

"A house requires an average of 5 kW of heat input to maintain an inside temperature of 20 ºC when the outside air is 0 ºC. (a) If the heating is accomplished by electrical resistance heaters, at 10 cents per kWh, how much does it cost to heat per day? (b) If an ideal heat pump were installed and operated as a Carnot engine [i.e., ] in reverse, what would the daily electricity bill be?"

(a) 5 kW24 hours$0.10=$12.00

(b) The so-called Coefficient of Performance for a heat pump is COP. For an idealized, lossless Carnot engine, the expression is . Here, the COP is K. The work necessary (back in heat/work mode again, not just temperature mode) to maintain the inside temperature of 20 ºC is kW. With the temperature differential involved here, a perfect heat pump can deliver the needed heat with a fraction of the work that it took the electrical resistance heaters: .34 kW24 hours$0.10=$0.82.