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CHAPTER 7
THE FIRST AND SECOND LAWS OF THERMODYNAMICS
7.1 The First Law of Thermodynamics, and Internal Energy
The First Law of thermodynamics is:
The increase of the internal energy of a system is equal to the sum of the heat added to the system plus the work done on the system.
In symbols:
dU = dQ + dW. 7.1.
You may regard this, according to taste, as any of the following
A fundamental law of nature of the most profound significance;
or A restatement of the law of conservation of energy, which you knew already;
or A recognition that heat is a form of energy.
or A definition of internal energy.
Note that some authors use the symbol E for internal energy. The majority seem to use U , so we shall use U here.
Note also that some authors write the first law as dU = dQ − dW , so you have to be clear what the author means by dW. A scientist is likely to be interested in what happens to a system when you do work on it, and is likely to define dW as the work done on the system, in which case dU = dQ + dW. An engineer, in the other hand, is more likely to be asking how much work can be done by the system, and so will prefer dW to mean the work done by the system, in which case dU = dQ − dW.
The internal energy of a system is made up of many components, any or all of which may be increased when you add heat to the system or do work on it. If the system is a gas, for example, the internal energy includes the translational, vibrational and rotational kinetic energies of the molecules. It also includes potential energy terms arising from the forces between the molecules, and it may also include excitational energy if the atoms are excited to energy levels above the ground state. It may be difficult to calculate the total internal energy, depending on which forms of energy you take into account. And of course the potential energy terms are always dependent on what state you define to have zero potential energy. Thus it is really impossible to define the total internal energy of a system uniquely. What the first law tells us is the increase in internal energy of a system when heat is added to it and work is done on it.
Note that internal energy is a function of state. This means, for example in the case of a gas, whose state is determined by its pressure, volume and temperature, that the internal energy is uniquely determined (apart from an arbitrary constant) by P, V and T – i.e. by the state of the gas. It also means that in going from one state to another (i.e. from one point in PVT space to another), the change in the internal energy is route-independent. The internal energy may be changed by performance of work or by addition of heat, or some combination of each, but, whatever combination of work and energy is added, the change in internal energy depends only upon the initial and final states. This means, mathematically, that dU is an exact differential (see Chapter 2, Section 2.1). The differentials dQ and dW , however, are not exact differentials.
Note that if work is done on a Body by forces in the Rest of the Universe, and heat is transferred to the Body from the Rest of the Universe (also known as the Surroundings of the Body), the internal energy of the Body increases by dQ + dW , while the internal energy of the Rest of the Universe
(the Surroundings) decreases by the same amount. Thus the internal energy of the Universe is constant. This is an equivalent statement of the First Law. It is also sometimes stated as “Energy can neither be created nor destroyed”.
7.2 Work
There are many ways in which you can do work on a system. You may compress a gas; you may magnetize some iron; you may charge a battery; you may stretch a wire, or twist it; you may stir a beaker of water.
Some of these processes are reversible ; others are irreversible or dissipative. The work done in compressing a gas is reversible if it is quasistatic, and the internal and external pressures differ from each other always by only an infinitesimal amount. Charging a lead-acid car battery may be almost reversible; charging or discharging a flashlight battery is not, because it has a high internal resistance, and the chemical reactions are irreversible. Stretching or twisting a wire is reversible as long as you do not exceed the elastic limit. If you do exceed the elastic limit, it will not return to its original length; that is, it exhibits elastic hysteresis. When you magnetize a metal sample, you are doing work on it by rotating the little magnetic moments inside the metal. Is this reversible? To answer this, read about the phenomenon of magnetic hysteresis in Chapter 12, Section 12.6, of Electricity and Magnetism.
Work that is reversible is sometimes called configuration work. It is also sometimes called PdV work , because that is a common example. Work that is not reversible is sometimes called dissipative work. Forcing an electric current through a wire is clearly dissipative.
For much of the time, we shall be considering the work that is done on a system by compressing it. Solids and liquids require huge pressures to change their volumes significantly, so we shall often be considering a gas. We imagine, for example, that we have a quantity of gas held in a cylinder by a piston. The work done in compressing it in a reversible process is − PdV. If you are asking yourself "Is P the pressure that the gas is exerting on the piston, or the pressure that the piston is exerting on the gas?", remember that we are considering a reversible and quasistatic process, so that the difference between the two is at all stages infinitesimal. Remember also that in calculus, if x is some scalar quantity, the expression dx doesn't mean vaguely the "change" in x (an ill-defined
dissipative part of the work done on the gas; it is unrecoverable as work, and is irretrievably converted to heat. You cannot use it to turn the paddle back. Nor can you cool the gas by turning the paddle backwards.
We can now define the increase of entropy in the irreversible process by TdS = dQ + dW irr ; that
is, dS dQ dW T
- (^) irr (^). In other words, since dW irr is irreversibly converted to heat, it is just as
though it were part of the addition of heat.
In summary, dU = dQ + dW and dU = TdS − PdV
apply whether there is reversible or irreversible work. But only if there is no irreversible (unrecoverable) work does dQ = TdS and dW = − PdV. If there is any irreversible work, dW = − PdV + dW irr and dQ = TdS − dW irr.
Of course there are other forms of reversible work than PdV work; we just use the expansion of gases as a convenient example.
Note that P , V and T are state variables (together, they define the state of the system) and U is a function of state. Thus the entropy , too, is a function of state. That is to say that the change in entropy as you go from one point in PVT -space to another point is route-independent. If you return to the same point that you started at (the same state, the same values of P , V and T ), there is no change in entropy, just as there is no change in internal energy.
Definition: The specific heat capacity C of a substance is the quantity of heat required to raise the temperature of unit mass of it by one degree. We shall return to the subject of heat capacity in Chapter 8. For the present, we just need to know what it means, in order to do the following exercise concerning entropy.
Exercise. A litre (mass = 1 kg) of water is heated from 0 oC to 100 oC. What is the increase of entropy? Assume that the specific heat capacity of water is C = 4184 J kg−^1 K−^1 , that it does not vary significantly through the temperature range of the question, and that the water does not expand significantly, so that no significant amount of work (reversible or irreversible) is done.
Solution. The heat required to heat a mass m of a substance through a temperature range dT is
mCdT. The entropy gained then is
mCdT T
. (^) The entropy gained over a finite temperature range is
therefore ln( 2 / 1 ) 1 4184 ln( 373. 15 / 273. 15 ) 1305 JK^1. 2 1
= = × × =^ −
∫ T mC T T
dT mC
T T
7.4 The Second Law of Thermodynamics
In a famous lecture entitled The Two Cultures given in 1959, the novelist C. P. Snow commented on a common intellectual attitude of the day - that true education consisted of familiarity with the humanities, literature, arts, music and classics, and that scientists were mere uncultured technicians and ignorant specialists who never read any of the great works of literature. He described how he had often been provoked by such an attitude into asking some of the self-proclaimed intellectuals if they could describe the Second Law of Thermodynamics – a question to which he invariably received a cold and negative response. Yet, he said, he was merely asking something of about the scientific equivalent of "Have you read a work of Shakespeare?"
So I suggest that, if you have never read a work of Shakespeare, take a break for a moment from thermodynamics, go and read A Midsummer Night's Dream , and come back refreshed and ready to complete your well-rounded education by learning the Second Law of Thermodynamics.
We have defined entropy in such a manner that if a quantity of heat dQ is added reversibly to a system at temperature T , the increase in the entropy of the system is dS = dQ / T. We also pointed out that if the heat is transferred irreversibly , dS > dQ / T.
Now consider the following situation (figure VII.1).
An isolated system consists of two bodies, A at temperature T 1 and B at temperature T 2 , such that T 2 > T 1. Heat will eventually be exchanged between the two bodies, and on the whole more heat will be transferred from B to A than from A to B. That is, there will be a net transference of heat, dQ , from B to A. Perhaps this heat is transferred by radiation. Each body is sending forth numerous photons of energy, but there is, on the whole, a net flow of photons from B to A. Or perhaps the two bodies are in contact, and heat is being transferred by conduction. The vibrations in the hot body are more vigorous than those in the cool body, so there will be a net transfer of heat from B to A. However, since the emission of photons in the first case, and the vibrations in the second place, are random, it will be admitted that it is not impossible that at some time more photons may move from A to B than from B to A. Or, in the case of conduction, most of the atoms in A happen to be moving to the right while only a few atoms in B are moving to the left in the course of their oscillations. But, while admitting that this is in principle possible and not outside
A B
T 1 T 2
dQ
FIGURE VII.
there is an increase in the entropy of the system as a whole. The principle of the increase of entropy applies to an isolated system.
In case you have ever wondered (who hasn’t?) how life arose on Earth, you now have a puzzle. Surely the genesis and subsequent evolution of life on Earth represents an increase in order and complexity, and hence a decrease in the entropy of mixing. You may conclude from this that the genesis and subsequent evolution of life on Earth requires Divine Intervention, or Intelligent Design, and that the Second Law of Thermodynamics provides Proof of the Existence of God. Or you may conclude that Earth is not an isolated thermodynamical system. Your choice.
