Something > Carnot efficiency ????? WTF?

[ironizer]

So I've made a very simple, ideal model of a heat engine, and my calculations just don't match up to the carnot thing: efficiency = 1-(Tc/Th)

 

It looks like this: I supposed a rectangular block, base of 1cm by 1cm, and a piston free to move up and down this block without friction. Also suppose that the walls are completely insulated thermally (no heat can transfer through, as to not screw up our calculations). Now we know that 1 atm = 1.03323 kg/sq cm, and the piston is 1x1cm so it must weigh 1.03323kg in order to apply a downward pressure of 1 atmosphere constantly. Assume a vacuum outside our piston, only gravity pushes the piston down.

 

So I've done 2 calculations, one with helium and one with chlorine. PV=nRT, P is 1, n is 1 because we calculate using one mole each, and R is constant. I know that the specific heat capacity for chlorine is 33.95 J/mol for each degree Kelvin. So I did this: suppose we apply 100 joules of heat INSIDE the rectangular cylinder, and the new temperature would be 100/33.95=2.945 degrees Kelvin increase. The 33.95 capacity says its for room temperature, so I calculated the volume of the gas at room temperature (R*(273+25)) and subtracted that from the new volume (after we applied our 100J heat) so it's R*(273+25+2.945) and I get a .24170839 Liters increase in volume, after applying just 100Joules of heat. If our rectangle is 1x1cm, it means that the height (distance the piston moves) is increased by 2.41708 meters, and we use work=F*D to find our increase in potential energy. Remember our piston weighs 1.03323 kg, so we do 1.03323*9.8*2.41708 = 24.47308 JOULES mechanical energy (potential). This is 24.47308% efficiency (we started with 100J)

 

WTF?! Did I screw up somewhere? I checked it over about 18 times and nothing looks wrong. But when we use the equation efficiency = 1-(Tc/Th) we get 1 - (273+25)/(273+25+2.945) = 0.009785 or 0.978% efficiency theoretically possible. 0.978% is a bit different than my 24.47308%.

 

I did the same thing for 1 mole of helium, and I got 39.97339%, because helium has lower specific heat capacity (20.786 J/mol K).

 

So carnot's theorem says that the efficiency is based solely on the difference in operating temperatures, I'm finding that my efficiency depends on the specific heat capacity and that's about it.

 

I know carnot's engine is a bit different, he does some weird crap, but my simple system can be made to cycle as well. I could have a "valve" that would let the temperature return to room temperature (273+25) as it was before we applied 100J, and then we could repeat the process.

 

I can't draw any lines. Is my stuff better than carnot's engine, or did I just fail at math?

 

Help! Thanks.