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- From: "WILLIAM JONES" <billj AT harborside.com>
- To: <homestead AT lists.ibiblio.org>
- Cc:
- Subject: [Homestead] Heat Engine: State Equation
- Date: Tue, 19 Jun 2012 06:18:30 GMT
It turns out to little more difficult than any word problem from the physics
text. The basic law of nature is Boyle's Gas law: PV = kNT, where P is
pressure, V is volume, N is the number of molecules of gas, T is the
temperature, and k is Boyle's Constant. A little thought will show that the
quantity conserved is the number of molecules N, since the bottle and bladder
and connecting tube form a closed system. Let's just ignore the volume of
the connecting tube for now in our first model. Solving for N yields N =
PV/k/T. When the gas is cold, and the air is all in the bottle, the pressure
just equals the ambient pressure Po. The volume of gas is 2 liters, which
we'll call Vo. And the temperature is just that of the cold reservoir Tw
(water temperature).
When the gas heats up, it occupies two containers, the bottle and the
bladder. The total number of gas molecules is the sum of these two
quantities. The number of molecules of gas in the bottle is given by
PVo/k/T, where P is the unknown pressure when heated, and T is the hot
temperature inside the bottle. The number of molecules in the bladder is
PV/k/Tw, where V is the unknown volume attained by the bladder.
Let's pause to appreciate the fact that Torricelli's Law allows us to use the
same P for both the bladder and the bottle. Also, note that Boyle's Constant
appears in the denominator of all three terms, allowing us to cancel through
on both sides. Finally we obtain
PVo/T + PV/Tw = PoVo/Tw
- [Homestead] Heat Engine: State Equation, WILLIAM JONES, 06/19/2012
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