4.1 Heat Engins and Second Law Statements
The First Law provides a constraint on the total energy contained in a system and its surroundings. If it disappears in one form from the system during any thermodynamic process of change, it must reappear in another form either within the system or in the surroundings. However, a pertinent question that one may often need to answer is: Is the process of change aimed at feasible? As may be evident, the first law provides no constraint on the possible direction a process may take place. Nevertheless, in the real world such constraints do exist. For example, heat always flows from a high temperature body to one at a lower temperature. Momentum flow is always prompted in the direction of a pressure gradient, and molecules always migrate from a region of higher to lower chemical potential. These observations clearly are indicative of the existence of a constraint on natural processes, which have never been found to be violated.

Further, it is common observation that work is readily transformed into other forms of energy, including heat.  But all efforts to develop a device that may work in a continuous manner and convert heat completely into work or any other form of energy have proved impossible. Experimental observations show that typically no more than 40-50% of the total heat available may be converted to work or other energy forms. This finding has led to the conclusion that heat is a lower form of energy in that while it may be feasible to “degrade” work to heat, it is impossible to “upgrade” heat completely into work.
Heat may be seen as a more primitive form of energy, as it always has to be made available from matter (say by combustion) and subsequently converted to work for carrying out activities useful to humans. In this sense one never derives work directly from the energy locked in matter as enthalpy. This prompts the natural question: what determines the efficiency of such a conversion of heat to work? Evidently one needs a limiting principle that may help answer this question. These considerations provide the basis for formulating the Second Law of Thermodynamics. 
2.1 Second Law Statements
It need be mentioned that the second law is a product of experiential observations involving heat engines that powered the Industrial Revolution of the 19th century. A heat engine is a machinethat produces work from heat through a cyclic process. An example is a steam power plant in which the working fluid (steam) periodically goes through a series of steps in a cyclic manner as follows: 
  • Step 1: Liquid water at ambient temperature is pumped into a boiler operating at high pressure
  • Step 2: Heat released by burning a fossil fuel is transferred in the boiler to the water, converting it to steam at high-temperature and pressure
  • Step 3: The energy contained in the steam is then transferred as shaft work to a turbine; during this process steam temperature and pressure are reduced.
  • Step 4: Steam exiting the turbine is converted to water by cooling it and transferring the heat released to the surroundings. The water is then returned to step 1. 
Like the steam power plant all heat engines absorb heat at a higher temperature body (source) and release a fraction of it to a low temperature body (sink), the difference between the two quantities constitutes the net work delivered during the cycle. The schematic of a heat engine (for example: steam / gas power plant, automotive engines, etc) is shown in fig. 4.1. As in the case of the steam cycle, a series of heat and work exchanges takes place, in each case a specific hot source and a cold sink are implicated. A schematic of such processes is suggested inside the     
Fig. 4.1 Schematic of Heat Engine


yellow circle between the hot and cold sources. The opposite of a heat engine is called the heat pump (refrigerators being an example of such device) is shown in fig. 4.2. There are indeed a large number other types of practical heat engines and power cycles. Select examples include: Ericsson Cycle,Stirling cycleOtto cycle (e.g. Gasoline/Petrol enginehigh-speed diesel engine), Diesel cycle (e.g. low-speed diesel engine), etc. The Rankine cycle most closely reproduces the functioning ofheat engines that use steam as the process fluid function (fig. 4.3); such heat engines are most commonly found in power generation plants. In such plants typically heat is derived from nuclear fission or the combustion of fossil fuels such ascoalnatural gas, and oil. Detailed thermodynamic analysis of the various heat engine cycles may be found in a number of textbooks (for example: J.W. Tester and M. Modell, Thermodynamics and its Applications, 3rd ed., Prentice Hall, 1999).
Fig. 4.2 Comparison of Heat Engine and Heat Pump


Fig. 4.3 Schematic of a Power Plant (Rankine) Cycle
As evident, the operation of practical heat engines requires two bodies at constant differential temperature levels. These bodies are termed heat reservoirs; they essentially are bodies with – theoretically speaking – infinite thermal mass (i.e., ) which therefore do not undergo a change of temperature due to either release or absorption of heat. The above considerations may be converted to a set of statements that are equivalent descriptors of the second law (R. Balzheiser, M. Samuels, and J. Eliassen, Chemical Engineering Thermodynamics, Prentice Hall, 1972):


Kelvin-Planck Statement:      It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work. 
Clausius Statement:   It is impossible to devise a cyclically operating device, the sole effect of which is to transfer energy in the form of heat from a low temperature body to a high temperature body.

4.2 Carnot Heat Engine Cycle and the 2nd Law
In theory we may say that a heat engine absorbs a quantity of heat from a high temperature reservoir at TH and rejects  amount of heat to a colder reservoir at TH. It follows that the net work W delivered by the engine is given by:
(4.1)
Hence the efficiency of the engine is:
(4.2)
(4.3)
(4.4)
Of the various forms of heat engines ideated, the Carnot engine proposed in 1824 by the French engineer Nicholas Leonard Sadi Carnot (1796-1832), provides a fundamental reference concept in the development of the second law. The so-called Carnot cycle (depicted in fig. 4.4) is a series of reversiblesteps executed as follows:


  • Step 1: A system at the temperature of a cold reservoir TC undergoes a reversible adiabatic compression which raises it temperature to that of a hot reservoir at TH.
  • Step 2: While in contact with the hot reservoir the system absorbs  amount of heat through an isothermal process during which its temperature remains at TH.  
  • Step 3: The system next undergoes a reversible adiabatic process in a direction reverse of step 1 during which its temperature drops back to  TC.
  • Step 4: reversible isothermal process of expansion at TC transfers amount of heat to the cold reservoir and the system state returns to that at the commencement of step 1.
Fig. 4.4 Carnot Cycle Processes
The Carnot engine, therefore, operates between two heat reservoirs in such a way that all heat exchanges with heat reservoirs occur under isothermal conditions for the system and at the temperatures corresponding to those of the reservoirs. This implies that the heat transfer occurs under infinitesimal temperature gradients across the system boundary, and hence these processes are reversible (see last paragraph of section 1.9). If in addition the isothermal and adiabatic processes are also carried out under mechanically reversible (quasi-static) conditions the cycle operates in a fully reversible manner. It follows that any other heat engine operating on a different cycle (between two heat reservoirs) must necessarily transfer heat across finite temperature differences and therefore cannot be thermally reversible. As we have argued in section that irreversibility also derives from the existence of dissipative forces in nature, which essentially leads to waste of useful energy in the conversion of work to heat. It follows therefore the Carnot cycle (which also comprises mechanically reversible processes) offers the maximum efficiency possible as defined by eqn. 4.3. This conclusion may also be proved more formally (see K. Denbigh, Principles of Chemical Equilibrium, 4th ed., Cambridge University Press, 1981).

We next derive an expression of Carnot cycle efficiency in terms of macroscopic state properties.  Consider that for the Carnot cycle shown in fig. 4.4 the process fluid in the engine is an ideal gas. Applying the eqn. 3.13 the heat interactions during the isothermal process may be shown to be: 
(4.4)
And 
Further for the adiabatic paths xy and zw using eqn. 3.17 one may easily derive the following equality:
(4.6)
Using eqns. 4.4 – 4.5 we may write: 
(4.7)
Finally applying eqn. (4.6) on obtains:
(4.8)
Hence, using eqn. 4.3: (4.9)
Eqn. (4.8) may be recast as:
(4.10)
As heat QH enters the system it is positive, while QC leaves the system, which makes it negative in value. Thus, removing the modulus, eqn. 4.10 may be written as:
(4.11)
Or: (4.12)
Or(4.13)
Consider now eqn. 4.9. For the Carnot efficiency to approach unity (i.e., 100%) the following conditions are needed:  Obviously neither situation are practicable, which suggests that the efficiency must always be less than unity. In practice, the naturally occurring bodies that approximate a cold reservoir are: atmospheres, rivers, oceans, etc, for which a representative temperature TC is ~ 3000K. The hot reservoirs, on the other hand are typically furnaces for which TH ~ 6000K. Thus the Carnot efficiency is ~ 0.5. However, in practice, due to mechanical irreversibilities associated with real processes heat engine efficiencies never exceed 40%


It is interesting to note that the final step of thermodynamic analysis of Carnot cycle (i.e., eqn. 4.13) leads to the conclusion that there exists a quantity  which add up to zero for the complete cycle. Let us explore extending the idea to any general reversible cycle (as illustrated in fig.4.5) run by any working fluid in the heat engine. 
Fig. 4.5 Illustration of an arbitrary cycle decomposed into a series of small Carnot cycles
The complete cycle may, in principle, be divided into a number of Carnot cycles (shown by dotted cycles) in series. Each such Carnot cycle would be situated between two heat reservoirs. In the limit that each cycle becomes infinitesimal in nature and so the number of such cycles infinity, the original, finite cycle is reproduced. Thus for each infinitesimal cycle the heat absorbed and released in a reversible manner by the system fluid may be written as and  respectively. Thus, now invoking eqn. 4.12 for each cycle:

Hence, applying eqn. 4.13 to sum up the effects of series of all the infinitesimal cycles, we arrive at the following relation for the entire original cycle:
(4.15)
Clearly, the relation expressed by eqn. 4.15 suggests the existence of a state variable of the form  as its sum over a cycle is zero. This state variable is termed as “Entropy” (S) such that:
(4.16)
Thus: 
Thus, for a reversible process: (4.17)
If applied to a perfectly reversible adiabatic process eqn. 4.16 leads to the following result: Thus, such a process is alternately termed as isentropic.


Since entropy is a state property (just as internal energy or enthalpy), even for irreversible process occurring between two states, the change in entropy would be given by eqn. 4.16. However since entropy is calculable directly by this equation one necessarily needs to construct a reversible process by which the system may transit between the same two states. Finite changes of entropy for irreversible processes cannot be calculated by a simple integration of eqn. 4.17. However, this difficulty is circumvented by applying the concept that regardless of the nature of the process, the entropy change is identical if the initial and final states are the same for each type of process. This is equally true for any change of state brought about by irreversible heat transfer due to finite temperature gradients across the system and the surroundings. The same consideration holds even for mechanically irreversibly processes. 

With the introduction of the definition of entropy the Carnot engine cycle may be redrawn on a temperature-entropy diagram as shown in fig. 4.6. 
Fig. 4.6 Representation of Carnot cycle on T-S diagram
(4.5)