Chapter 3: The Second Law of Thermodynamics | Lectures on Thermodynamics and Statistical Mechanics

3. The Second Law of Thermodynamics

3.1 Carnot cycle

A thermodynamic engine operates by taking in heat from a hot reservoir and performing certain work and then giving up a certain amount of heat into a colder reservoir. If it can be operated in reverse, it can function as a refrigerator. The Carnot cycle is a reversible cyclic process (or engine) made of the following four steps:

It starts with an adiabatic process which raises the temperature of the working material of the engine to, say,
This is followed by a isothermal process, taking in heat from the reservoir at .
The next step is an adiabatic process which does some amount of work and lowers the temperature of the material to .
The final step is isothermal, at the lower temperature , dumping some amount of heat into a colder reservoir, with the material returning to the thermodynamic state at the beginning of the cycle.

This is an idealized engine, no real engine can be perfectly reversible. The utility of the Carnot engine is to give the framework and logic of the arguments related to the second law of thermodynamics. We may say it is a gedanken engine. The processes involved in the Carnot cycle may refer to compression and expansion if the material is a gas; in this case, the cycle can be illustrated in a diagram as shown in Fig. 3.1. But any other pair of thermodynamic variables will do as well. We can think of a Carnot cycle utilizing magnetization and magnetic field, or surface tension and area, or one could consider an electrochemical cell.

Let the amount of heat taken in at temperature be and let the amount of heat given up at the lower temperature be . Since this is an idealized case, we assume there is no loss of heat due to anything like friction. Thus the amount of work done, according to the first law is

(3.1)

The efficiency of the engine is given by the amount of work done when a given amount of heat is supplied, which is . (The heat which is dumped into the reservoir at lower temperature is not usable for work.) The efficiency for a Carnot cycle is thus

(3.2)

The importance of the Carnot cycle is due to its idealized nature of having no losses and because it is reversible. This immediately leads to some simple but profound consequences.

3.2 The second law

The second law of thermodynamics is a statement of what we know by direct experience. It is not something that is derived from more fundamental principles, even though a better understanding of this law has emerged over time. There are several ways to state the second law, the most common ones being the Kelvin statement and the Clausius statement.

K: Kelvin statement of the second law: There exists no thermodynamics process whose sole effect is to extract an amount of heat from a source and convert it entirely to work.
C: Clausius statement of the second law: There exists no thermodynamics process whose sole effect is to extract an amount of heat from a colder source and deliver it to a hotter source.

The key word here is “sole". Consider the expansion of a gas and consequent conversion of heat into work. Complete conversion can be achieved but this is not the sole effect, for the state of the system has been changed. The second law does not forbid such a process.

The two statements are equivalent. This can be seen by showing that and , where the tildes denote the negation of the statements.

First consider . Consider two heat reservoirs at temperatures and , with . Since K is false, we can extract a certain amount of heat from the colder reservoir (at ) and convert it entirely to work. Then we can use this work to deliver a certain amount of heat to the hotter reservoir. For example, work can be converted to heat by processes like friction. So we can have some mechanism like this to heat up the hotter source further. The net result of this operation is to extract a certain amount of heat from a colder source and deliver it to hotter source, thus contradicting . Thus .

Now consider . Since is presumed false, we can extract an amount of heat, say Q2, from the colder reservoir (at ) and deliver it to the hotter source. Then we can have a thermodynamic engine take this amount of heat from the hotter reservoir and do a certain amount of work delivering an amount of heat to the colder reservoir. The net result of this cycle is to take the net amount of heat from the reservoir at TL and convert it entirely to work. This shows that .

The two statements and show the equivalence of the Kelvin and Clausius statements of the second law.

The second law is a statement of experience. Most of the thermodynamic results can be derived from a finer description of materials, in terms of molecules, atoms, etc. However, to date, there is no clear derivation of the second law. Many derivations, such as Boltzmann’s -theorem, or descriptions in terms of information, have been suggested, which are important in their own ways, but all of them have some additional assumptions built in. This is not to say that they are not useful. The assumptions made have a more fundamental nature, and do clarify many aspects of the second law.

3.3 Consequences of the second law

Once we take the second law as an axiom of thermodynamics, there are some important and immediate consequences. The first result is about the efficiency of the Carnot cycle, captured as the following theorem.

The proof is easy, based on the second law. Consider two engines, say a Carnot engine and another engine we call , and two reservoirs, at temperatures and . The Carnot engine is reversible, so we can operate it as a refrigerator. So we can arrange for it to take a certain amount of heat from the colder reservoir and deliver an amount to the hotter reservoir. This will, of course, require work to drive the Carnot engine. Now we can arrange for to take from the hotter reservoir, and do an amount of work , delivering heat to the colder reservoir. The efficiencies are given by

(3.3)

Assume is more efficient. Then , or . Thus The net amount of heat extracted from the hotter reservoir is zero, the net amount of heat extracted from the colder reservoir is . This is entirely converted to work (equal to ) contradicting the Kelvin statement of the second law. Hence our assumption of must be false, proving the Carnot theorem. Thus we must have

We also have an immediate corollary to the theorem:

Proposition 1 All perfectly reversible engines operating between two given temperatures have the same efficiency.

This is also easily proved. Consider the engine to be a Carnot engine. From what we have already shown, we will have . Since E2 is reversible, we can change the roles of and , running as a refrigerator and as the engine producing work. In this case, the previous argument would lead to . We end up with two statements, η1 ≤ η2 and η2 ≤ η1. The only solution is η1 = η2. Notice that this applies to any reversible engine, since we have not used any specific properties of the Carnot engine except reversibility.

If an engine is irreversible, the previous arguments hold, showing η2 ≤ η1, but we cannot get the other inequality because E2 is not reversible. Thus irreversible engines are less efficient than the Carnot engine.

A second corollary to the theorem is the following:

Proposition 2 The efficiency of a Carnot engine is independent of the working material of the engine.

The arguments so far did not use any specific properties of the material of the Carnot engine, and since all Carnot engines between two given reservoirs have the same efficiency, this clear. We now state another important consequence of the second law.

Proposition 3 The adiabatics of a thermodynamic system do not intersect.

We prove again by reductio ad absurdum. Assume the adiabatics can intersect, as shown in Fig. 3.2. Then we can consider a process going from A to B which is adiabatic and hence no heat is absorbed or given up, then a process from B to C which absorbs some heat ∆Q, and then goes back to A along another adiabatic. Since the thermodynamic state at A is restored, the temperature and internal energy are the same at the end as at the beginning, so that ∆U = 0. Thus by the first law, ∆Q = ∆W , which means that a certain amount of heat is

absorbed and converted entirely to work with no other change in the system. This contradicts the Kelvin statement of the second law. It follows that adiabatics cannot intersect.

3.4 Absolute temperature and entropy

Another consequence of the second law is the existence of an absolute temperature. Although we have used the notion of absolute temperature, it was not proven. Now we can show this just from the laws of thermodynamics.

We have already seen that the efficiency of a Carnot cycle is given by

(3.4)

where is the amount of heat taken from the hotter reservoir and is the amount given up to the colder reservoir. The efficiency is independent of the material and is purely a function of the lower and upper temperatures. The system under consideration can be taken to be in thermal contact with the reservoirs, which may be considered very large. There is no exchange of particles or any other physical quantity between the reservoirs and the system, so no parameter other than the temperature can play a role in this. Let denote the empirically defined temperature, with and corresponding to the reservoirs between which the engine is operating. We may thus write

(3.5)

for some function of the temperatures. Now consider another Carnot engine operating between and , with corresponding ’s, so that we have

(3.6)

Now we can couple the two engines and run it together as a single engine, operating between and , with

(3.7)

Evidently

(3.8)

so that we get the relation

(3.9)

This requires that the function must be of the form

(3.10)

for some function . Thus there must exist some function of the empirical temperature which can be defined independently of the material. This temperature is called the absolute temperature. Notice that since , we have if . Thus |f| should be an increasing function of the empirical temperature. Further we cannot have for some temperature . This would require . The corresponding engine would take some heat from the hotter reservoir and convert it entirely to work, contradicting the Kelvin statement. This means that we must take f to be either always positive or always negative, for all θ. Conventionally we take this to be positive. The specific form of the function determines the scale of temperature. The simplest is to take a linear function of the empirical temperatures (as defined by conventional thermometers). Today, we take this to be from the hotter reservoir and convert it entirely to work, contradicting the Kelvin statement. This means that we must take f to be either always positive or always negative, for all . Conventionally we take this to be positive. The specific form of the function determines the scale of temperature. The simplest is to take a linear function of the empirical temperatures (as defined by conventional thermometers). Today, we take this to be

(3.11)

The unit of absolute temperature is the kelvin.

Once the notion of absolute temperature has been defined, we can simplify the formula for the efficiency of the Carnot engine as

(3.12)

Also we have , which we may rewrite as . Since Q2 is the heat absorbed and is the heat released into the reservoir, we can assign signs to the , for intake and - for release of heat, and write this equation as

(3.13)

In other words, if we sum over various steps (denoted by the index ) of the cycle, with appropriate algebraic signs,

(3.14)

If we consider any closed and reversible cycle, as shown in Fig. 3.3, we can divide it into small cycles, each of which is a Carnot cycle. A few of these smaller cycles are shown by dotted lines, say with the long dotted lines being adiabatics and the short dotted lines being isothermals. By taking finer and finer such divisions, the error in approximating the cycle by a series of closed Carnot cycles will go to zero as the number of Carnot cycles goes to infinity. Since along the adiabatics, the change in is zero, we can use the result (3.14) above to write

(3.15)

where we denote the heat absorbed or given up during each infinitesimal step as . The statement in equation (3.15) is due to Clausius. The important thing is that this applies to any closed curve in the space of thermodynamic variables, provided the process is reversible.

This equation has another very important consequence. If the integral of a differential around any closed curve is zero, then we can write the differential as the derivative of some function. Thus there must exist a function such that

(3.16)

This function is called entropy. It is a function of state, given in terms of the thermodynamic variables.

Clausius’ inequality

Clausius’ discovery of entropy is one of most important advances in the physics of material systems. For a reversible process, we have the result,

(3.17)

as we have already seen. There is a further refinement we can make by considering irreversible processes. There are many processes such as diffusion which are not reversible. For such a process, we cannot write . Nevertheless, since entropy is a function of the state of the system, we can still define entropy for each state. For an irreversible process, the heat transferred to a system is less than where is the entropy change produced by the irreversible process. This is easily seen from the second law. For assume that a certain amount of heat is absorbed by the system in the irreversible process. Consider then a combined process where the system changes from state to state in an irreversible manner and then we restore state by a reversible process. For the latter step . The combination thus absorbs an amount of heat equal to with no change of state. If this is positive, this must be entirely converted to work. However, that would violate the second law. Hence we should have

(3.18)

If we have a cyclic process, since is a state function, and hence

(3.19)

with equality holding for a reversible process. This is known as Clausius’ inequality. For a system in thermal isolation, , and the condition becomes

(3.20)

In other words, the entropy of a system left to itself can only increase, equilibrium being achieved when the entropy (for the specified values of internal energy, number of particles, etc.) is a maximum.

The second law has been used to define entropy. But once we have introduced the notion of entropy, the second law is equivalent to the statement that entropy tends to increase. For any process, we can say that

(3.21)

We can actually see that this is equivalent to the Kelvin statement of the second law as follows. Consider a system which takes up heat at temperature . For the system (labeled 1) together with the heat source (labeled 2), we have . But the source is losing heat at temperature and if this is reversible, . Further if there is no other change in the system, and . Thus

(3.22)

Since and this equation implies that the work done by the system cannot be positive, if we have . Thus we have arrived at the Kelvin statement that a system cannot absorb heat from a source and convert it entirely to work without any other change. We may thus restate the second law in the form:

Proposition 4 Second law of thermodynamics: The entropy of a system left to itself will tend to increase to a maximum value compatible with the specified values of internal energy, particle number, etc.

Nature of heat flow

We can easily see that heat by itself flows from a hotter body to a cooler body. This may seem obvious, but is a crucial result of the second law. In some ways, the second law is the formalization of such statements which are “obvious" from our experience.

Consider two bodies at temperatures and , thermally isolated from the rest of the universe but in mutual thermal contact. The second law tells us that . This means that

(3.23)

Because the bodies are isolated from the rest of the world, , so that we can write the condition above as

(3.24)

If ,we must have and if . Either way, heat flows from the hotter body to the colder body.

3.5 Some other thermodynamic engines

We will now consider some other thermodynamic engines which are commonly used.

Otto cycle

The automobile engine operates in four steps, with the injection of the fuel-air mixture into the cylinder. It undergoes compression which can be idealized as being adiabatic. The ignition then raises the pressure to a high value with almost no change of volume. The high pressure mixture rapidly expands, which is again almost adiabatic. This is the power stroke driving the piston down, The final step is the exhaust when the spent fuel is removed from the cylinder. This step happens without much change of volume. This process is shown in Fig. 3.4. We will calculate the efficiency of the engine, taking the working material to be an ideal gas.

Heat is taken in during the ignition cycle to . The heat comes from the chemical process of burning but we can regard it as heat taken in from a reservoir. Since this is at constant volume, we have

(3.25)

Heat is given out during the exhaust process to , again at constant volume, so

(3.26)

Further, states and are connected by an adiabatic process, so are and . Thus

is preserved for these processes. Also, , so we have

(3.27)

This gives

(3.28)

The efficiency is then

(3.29)

Diesel cycle

The idealized operation of a diesel engine is shown in Fig. 3.5. Initially only air is admitted into the cylinder. It is then compressed adiabatically to very high pressure (and hence very high temperature). Fuel is then injected into the cylinder. The temperature in the cylinder is high enough to ignite the fuel. The injection of the fuel is controlled so that the burning happens at essentially constant pressure ( to in figure). This is the key difference with the automobile engine. At the end of the burning process the expansion continues adiabatically (part to ). From back to we have the exhaust cycle as in the automobile engine.

Taking the air (and fuel) to be an ideal gas, we can calculate the efficiency of the diesel engine. Heat intake (from burning fuel) is at constant pressure, so that

(3.30)