Chapter 6 The Second Law of Thermodynamics 6

Chapter 6 The Second Law of Thermodynamics

6. 1 Introduction The first law of thermodynamics is simple, general, but does not constitute a complete theory because certain processes it permits do not occur in nature!

The problems arise from: 1. The first law of thermodynamics focuses on states of equilibrium. 2. The exact process by which a system reaches the final state from its initial state is immaterial. i. e. the transition is independent of the particular path taken 3. The theory emphasizes reversible processes! Yet, real processes are irreversible!

Examples of processes which are not prohibited by the first law, but cannot happen in a real world. • Perfect machine • Transfer heat from cold to hot subjects • Gas expansion 6. 2 The mathematical concept of entropy

The reciprocal of the absolute temperature is an integrating factor that permits the replacement of the inexact differential by the exact differential . The above equation is the Clausius definition of the entropy S. The first law of thermodynamics can now be expressed as for a reversible process

Pressure P can also act as an integration factor to let the inexact differential be replaced by the exact differential dv i. e.

6. 3 Irreversible Processes 1. A battery discharges through a resistor, releasing energy. The reverse process will not occur. 2. Two gases, initially in separated adjoining chambers, will mix uniformly. 3. A free expansion of gas (in Gay-Lussar-Joule experiment) 4. Heat flows from a high temperature body to a low temperature reservoir in the absence of other effect

Two statements of the second law of thermodynamics: Clausius Statement: It is impossible to construct a device that operates in a cycle and whose sole effect is to transfer heat from a cooler body to a hotter body. Kevin-Planck Statement: It is impossible to construct a device that operates in a cycle and produces no other effects than the performance of work and the exchange of heat with a single reservoir.

6. 4 Carnot’s Theorem When assuming should be smaller then since If then

As a result, the device does no work, but extracts heat from the cold reservoir and delivers it to the hot reservoir. Such a conclusion is again Clausius statement. 6. 5 The Clausius Inequality & the Second Law

Clausius Inequality For an irreversible process connecting states 1 and 2

Thus or The entropy of an isolated system increases in any irreversible process and is unaltered in any reversible process. This is the principle of increasing entropy. The fact that the entropy of an isolated system can never decrease in a process provides a direction for the sequence of natural events.

6. 6 Entropy and Available Energy It is impossible to utilize all the internal energy of a body for the production of mechanical work. There exists no process that can increase the available energy in the universe. 6. 7 Absolute Temperature 6. 8 Combined First & Second Laws For a reversible process

The second law states that Thus Comparing with the general expression One gets

7. 1 Entropy Changes in Reversible Processes • The following special cases will be examined first: 1. 2. 3. 4. 5. Adiabatic process: dq = 0 and ds =0. A reversible adiabatic process has a constant entropy (i. e. isentropic). Isothermal process: Tds = dq, the integrated solution is ∆s = q/T Isothermal and isobaric change of phase: ∆s = l/T. is the latent heat of transformation. Isochoric process: Since Tds = dq = du + Pd. V, then Tds = du =cv d. T, then ∆s = cv ln(T 2/T 1) if cv is constant. Isobaric process: Tds = dq = dh – vd. P ; Tds = cpd. T; If cpis constant, ∆s = cp ln(T 2/T 1)

7. 2 Temperature-Entropy Diagrams • The total amount of heat transferred in a reversible process can be obtained from dq= Tds. • The integrated solution of dq = Tds corresponds to the area under a curve in a T-s diagram. • For the Carnot cycle, the T-s diagram looks like

7. 3 Entropy Change of the Surroundings for a Reversible Process • In any reversible process, the entropy change of the Universe is Always zero. • When there is a reversible flow of heat between a system and its surroundings, the temperatures of the system and the surroundings are essentially equal. • The heat flow out of the surroundings will be equal to the heat flow into the system. • dssystem + dssurroundings = dsuniverse

7. 4 Entropy Change for an Idea Gas • • For a reversible process: Tds = dq = du + pdv, For ideal gas: du = cvd. T Thus, ds = (cv/T)d. T + (p/T)dv = (cv/T)d. T + Rdv Since T and v are independent variables, integration can be carried out separately, yielding s 2 -s 1 = cv ln(T 2/T 1) - Rln(v 2/v 1). • The above relationship explains why temperature decreases during an isentropic expansion.

7. 5 The Tds equations

7. 6 Entropy Change in Irreversible Processes • It is not possible to calculate the entropy change ΔS = SB - SA for an irreversible process between A and B, by integrating dq / T, the ratio of the heat increment over the temperature, along the actual irreversible path A-B characterizing the process. • However, since the entropy is a state function, the entropy change ΔS does not depend on the path chosen. The calculation of an irreversible process can be carried out via transferring the process into many reversible ones: • Three examples will be discussed here: (1) heat exchange between two metal blocks with different temperatures; (2) Water cooling from 90 to a room temperature; (2) A falling object.

7. 7 Free Expansion of an Ideal Gas

7. 8 Entropy Change for a Liquid or Solid •