The third exam will be next Tuesday, 23 Nov. The exam will cover chapters 9 through 12. Chapter 13 will not be included on the exam.
One of the earliest and most important applications of thermodynamics is the understanding of engines. For our purposes, a heat engine is a system that goes through changes in pressure, volume, and temperature, in a cyclic fashion. That is, the system repeatedly follows the same changes in P, V, and T, returning to its starting conditions to begin another cycle.
By far the most common engines involve a system consisting of a gas,
so this is what we will concentrate on. We can represent the state
of the gas on a PV diagram. (Since, for a given quantity of an ideal
gas, T = PV/nR, specifying PV fixes T.) A cyclic process traces out
a closed curve on the PV diagram. (diagram) The net work done by the
engine per cycle equals the area enclosed by the curve,
Wcycle = Area enclosed in PV curve.
A heat engine can be represented schematically as a system connected to a hot reservoir and a cold reservoir. The system absorbs an amount of heat Qh from the hot reservoir, gives off an amount of heat Qc to the cold reservoir, and produces an amount of work W on each cycle. Since the thermodynamic process of the engine is cyclic, it returns to its starting condition after each cycle. Being in the same condition means that the internal energy of the system is the same, or DUcycle = 0 for a complete cycle.
The first law tells us that DUcycle = Qcycle - Wcycle, or, since DUcycle = 0, Qcycle = Qh - Qc = Wcycle. This is just conservation of energy. The system takes in an amount of heat energy Qh, and gives off an amount of heat energy Qc and work Wcycle.
The efficiency, e, of the engine is defined as the fraction of the
heat energy taken in that is converted to work:
e = W/Qh = (Qh - Qc)/Qh
= 1 - Qh/Qc. If the engine converts all the
input energy to heat (W = Qh) then the efficiency is
100%. If no work is produced then the efficiency is 0.
As applied to heat engines, the second law of thermodynamics says that the amount of work produced by a heat engine is always less than the amount of input energy, or, it is impossible for an engine to be 100% efficient. Mathematically, W < Qh.
The Carnot engine is a theoretical device that is useful because it
was shown to be the most efficient engine operating between a hot
and cold reservoirs at temperatures Th and Tc,
allowed by the laws of Physics. From a study of the Carnot engine,
we can deduce the maximum efficiency for an engine, only
knowing the hot and cold temperatures at which heat is transferred.
The result is:
ec = 1 - (Tc/Th)
where, of course, the temperatures must be in kelvin.
Notice that for the Carnot efficiency to equal 1, the cold temperature must be zero kelvin -- absolute zero. Since it is impossible to get anything to a temperature of absolute zero, it is impossible to make an engine that is 100% efficient. Because most exhaust goes to the atmosphere at a temperature of about 300°K, the usual route to higher efficiency is to raise Th for the engine. But this runs into practical difficulties such as the ability of materials to stand up to large temperature swings.
A heat engine performs 200 J of work in each cycle and has an efficiency of 30%. For each cycle of operation, (a) how much heat is absorbed and (b) how much heat is expelled?
Using e = W/Qh, we deduce that the heat absorbed is
Qh = W/e = 200J/0.3 = 667 J.
And since W = Qh - Qc, we find that
Qc = Qh - W = 667 J - 200 J = 467 J.
In one cycle, a heat engine absorbs 500 J from the high temperature reservoir and expels 300 J to a low temperature reservoir. If the efficiency of this engine is 60% of the efficiency of a Carnot engine, what is the ratio of the low temperature to the high temperature in the Carnot engine?
First, the efficiency of this engine is e = 1 -
(Qc/Qh) = 1 - (300/500) = 0.40 = 40%. If this
is 60% of the Carnot efficiency, then e = 60% ec, or
ec = e/0.6 = 0.4/0.6 = 0.67 = 67%. Finally, to get the
ratio of the low to high temperatures, use the relation
ec = 1 - (Tc/Th), so that
(Tc/Th) = 1 - e = 1 - 0.67 = 0.33.
Another result from the study of heat engines is the concept of entropy. Entropy is a measure of the amount of disorder in a system. For example, the internal structure of ice is more ordered than that of water, which is again more ordered than steam. The entropy of ice is less than the entropy of water is less than the entropy of steam (for a fixed number of H2O molecules).
Entropy is represented by the letter S. In our discussions, we will
only deal with changes in entropy, in the same way that we only deal
with changes in internal energy. "The change in entropy, DS, between two equilibrium states is given by the
heat transferred, DQ, divided by the
absolute temperature, T, of the system in this interval":
DS = DQ/T.
We can restate the second law of thermodynamics in terms of entropy: the entropy of the universe increases in all natural processes. This does not mean that entropy is always increasing everywhere. The entropy of a small system can be decreased, but the second law says that there will be a corresponding increase in the entropy outside of the system which exceeds the decrease in entropy of the system.
Calculate the change in entropy when 300 g of lead melts at 327°C (600°K). Lead has a latent heat of fusion of 2.45x104J/kg.
Let's determine the amount of heat absorbed to melt the lead, then
use DS = DQ/T to
determine the change in entropy.
Q = mLf = (0.30kg)(2.45x104J/kg) =
7.35x103J.
Thus
DS = (7.35x103J)/(600°K) =
12.3J/K.
As strange as this may at first seem, we begin our discussion of
vibrations and waves by introducing a force. This force, known as
the spring force, we briefly discussed in the context of potential
energy, and elasticity. Not just springs, but many solids, exhibit a
force that is proportional to a displacement:
Fs = -kx
where x is the displacement from equilibrium, and k is called the
spring constant, in N/m.
The minus sign means that the force acts against the displacement, trying to push or pull the object back to the equilibrium position, hence the term "restoring force" is sometimes used. The force varies linearly with the displacement, that is, if the displacement doubles, so does the force; if the displacement is halved, the force is halved. Finally, the value of k is a measure of stiffness. Stiff springs have a large value of k, soft springs have a small value of k. The linear, restoring behavior of the spring force is known as Hooke's law.
Now imagine a mass attached to one end of a spring, with the other end of the spring attached to a wall. The mass is free to slide on a frictionless horizontal surface. If the mass is pushed a bit, compressing the spring, then released, what will happen? The spring will push back against the mass, accelerating it towards the equilibrium position (x=0). As the mass accelerates, it will gain speed, until, reaching x=0, the force vanishes.
But the mass will continue moving (Newton's first law), passing x=0 and continuing on, stretching the spring. As the spring stretches, there is again a restoring force on the mass, pulling back towards x=0. Now the force will slow down the mass, causing it to stop, then accelerating it back towards x=0. The mass will reach the equilibrium position, and continue moving past, back to its starting location, and the cycle begins again.
Repetitive or cyclic motion such as this is called harmonic motion. When the force responsible for the motion is a linear restoring force (Hooke's law type force), then the motion is called simple harmonic.
For the example given above, Newton's second law is:
ma = -kx or a = -(k/m)x
From this we see that the acceleration is zero at the equilibrium
position (x=0), and the acceleration is maximum when x is maximum.
The maximum displacement of the mass from the equilibrium position
is called the amplitude of the motion, A. The maximum
acceleration occurs for x = +/-A.
In chapter 5 we saw that the potential energy stored in a spring
is:
PEs = (1/2)kx2, and we saw how to use this
with conservation of energy to solve problems. By applying this to
the case of a mass attached to a spring, executing simple harmonic
motion, we can find a relation between the velocity and position of
the mass.