Chapter 2: Entropy PDF

# Chapter 2: Entropy ## 2.1 Concept of Entropy - A thermodynamic variable which is a function of the state of the system called entropy. - It will have different values at the initial and final state of a thermodynamic process. - It is a measurable property of the system. - It describes the abi...

# Chapter 2: Entropy ## 2.1 Concept of Entropy - A thermodynamic variable which is a function of the state of the system called entropy. - It will have different values at the initial and final state of a thermodynamic process. - It is a measurable property of the system. - It describes the ability of the system to do work. - Entropy is very useful in the study of the behavior of a heat engine. ## 2.2 Clausius Theorem (or Change of Entropy in a Reversible Process) **Statement**: Clausius theorem states that in any reversible cycle, the net change in entropy is zero. **Proof**: This can be proven for the Carnot cycle (reversible). A Carnot cycle has two isothermal and two adiabatic processes. - In an isothermal expansion, the working substance takes Q1 amount of heat from the source at constant temperature T1. This increases the working substance's entropy by (Q1/T1). - During adiabatic expansion, no heat is taken in or given up, so there's no change in entropy. - During isothermal compression the working substance gives up Q2 amount of heat to the sink at constant temperature T2. This decreases the working substance's entropy by (Q2/T2). - During adiabatic compression, there is no change in entropy. Therefore, the total change of entropy is 0. **In general**: In a reversible process, the net change in entropy is zero. ## 2.3 Change of Entropy in an Irreversible Process If the working substance in an engine performs an irreversible cycle, absorbing Q1 amount of heat at temperature T1 from the source and rejecting Q2 amount of heat to the sink at temperature T2, then: - The efficiency of this irreversible cycle is less than a reversible engine working between the same two temperatures T1 and T2, and the heat engine's efficiency is less than a reversible engine, meaning (Q2/T2) is greater than (Q1/T1). - The source loses entropy by (Q1/T1) and the sink gains entropy by (Q2/T2). - The net change in entropy of the working substance is zero because when the cycle is completed, the working substance returns to its initial state. - The net change in entropy of the system in the irreversible cycle is: (Q2/T2) - (Q1/T1), which is greater than zero. Therefore, entropy of a system increases during an irreversible process. ## 2.4 Second Law of Thermodynamics in terms of Entropy - Let S1 and S2 be the entropies of the initial and final states of a system respectively. The change in entropy of the system is given by: ``` S2 - S1 = ∫ (dQ/T) ``` - This relation only applies for a reversible path. - If the initial and final states are infinitely close to each other, the above relation can be written as: ``` dS = dQ/T ``` - This is the second law of thermodynamics in mathematical form. dQ is the amount of heat absorbed or rejected at temperature T. - The principle of entropy is: - dQ/T >= 0 or dS >= 0 - This embodies both Kelvin-Planck and Clausius statements of the second law of thermodynamics. - According to Kelvin-Planck's statement, a perfect heat engine cannot convert all the amount of heat Q taken from the source at temperature T into work. If such a heat engine exists, then decrease of entropy of the source (dQ/T) = 0. - Since the working substance will return to the initial state after completing the cycle, hence entropy of the working substance and surrounding decrease. - This is against the principle of increase of entropy. - According to Clausius statement, a perfect refrigerator which is unaided by any external agency, cannot exist. A perfect refrigerator transfers Q amount of heat from a cold body at temperature T2 to a hot body at temperature T1: - Decrease in entropy of cold body = Q/T2. - Increase in entropy of hot body = Q/T1. - Entropy of the working substance remains unchanged. - Total entropy of the system and surroundings = (Q/T1) - (Q/T2) - Since T1 > T2, total entropy of the system and surroundings will decrease, which is against the principle of entropy because the entropy of a thermodynamic system plus surroundings either remains constant or increases but can never decrease. Therefore, the principle of increase of entropy is consistent with Kelvin-Planck and Clausius Statements of the second law of thermodynamics. ## 2.5 Entropy of a Perfect Gas Let 1 gm of a perfect gas at temperature T. - Cv = Specific heat of gas at constant volume - Cp = Specific heat of gas at constant pressure - r = Gas constant for 1 gm of gas - J = Joule's constant. If dQ is the amount of heat given to the gas at temperature T in a reversible process, then the change in entropy of the gas is: ``` dS = dQ/T ``` - According to the first law of thermodynamics: ``` dQ = dU + dW ``` where: - dU = increase in internal energy - dW = work done against external pressure - dW = PdV/J ## 2.6 Entropy as a Measure of Disorder - The term entropy (S) was introduced by R.J.E. Clausius in 1854. - It is the extent of disorder or randomness in a system. - It is a property of a substance which measures the *disorder or randomness* in a system. - If the value of a parameter remains unchanged on partition of the system, it is called a *intensive parameter*. - If the value of a parameter remains unchanged on partition of the system, it is called a *extensive parameter*. ## 2.10 Third Law of Thermodynamics - The change of entropy is given by: ``` dS = dQ/T ``` - Where: - dQ = small amount of heat supplied to the system. - T = Absolute temperature of the system. - According to sign conventions, heat gained by the system is positive and heat lost is negative. - Entropy increases when heat is added to a system and decreases when heat is taken out. ## 2.11 Reversible Process - A process in which the system exactly passes through the same intermediate states in a such a way that the system can also be retraced in the opposite direction and the system returns to its initial state, is called a reversible process. - A process is reversible if: - It takes place at an infinitely slow rate. - The working substance loses no heat by conduction, convection or radiation. - The work is done to overcome friction. - No diffusion of fluids. - No conduction or convection processes. - No generation of irreversible heat, for example, by the passage of an electric current. ## 2.13 Law of Increase of Entropy - Consider an isolated dynamic system consisting of a large number of particles where the particles of the system can interact with one another, but not with the surroundings. There is no exchange of energy (heat) of the system with the surroundings. - When all the parts of the system are at the same temperature, the system is said to be in *thermal equilibrium*. - The system will be in *maximum probability* (the state of maximum probability has negligible probability). - Entropy of the system is given by: ``` S = k log W ``` This means that as W increases, S will also increase. When a system comes back to the state of maximum probability, W will become maximum, which means that S will become maximum and constant. - Therefore, in *thermal equilibrium*, the change in entropy is zero (AS = 0). - If a system moves from an *inequilibrium* to an *equilibrium state*, the thermodynamic probability will increase, which will also increase the entropy. - Once a system reaches the state of equilibrium, it cannot go back to the state of inequilibrium. This is an *irreversible process* and the change in entropy will always be greater than zero (AS > 0). Therefore, entropy always increases in natural processes which are all irreversible process. ## 2.14 Examples of Increase of Entropy - 1. Transfer of Heat and the Second Law of Thermodynamics - 2. Expansion of a Gas ## 2.17 Temperature-Entropy (T-S) Diagram - Carnot's reversible cycle consists of four steps: 1. Isothermal Expansion: Work done by the system is equal to the heat absorbed from the source (Q1). Entropy increases. 2. Adiabatic Expansion: Work done by the system is equal to the decrease in internal energy of the system. The entropy remains constant. 3. Isothermal Compression: Work done on the system is equal to the heat released to the sink (Q2). Entropy decreases. 4. Adiabatic Compression: Work done on the system is equal to the increase in internal energy of the system. The entropy remains constant. - The net change in entropy of the working substance in one complete Carnot's cycle is zero. - The efficiency of Carnot's heat engine is given by: ``` η = (Q1 - Q2)/ Q1 = (T1 - T2) / T1 ``` - The efficiency of Carnot's heat engine is the same as obtained on the P-V diagram. ## 2.18 Unattainability of Absolute Zero - The unavailability of absolute zero can be demonstrated via the *adiabatic demagnetization* process. - This process cannot reach absolute zero because of the constraint of Nernst's theorem. - According to Nernst's theorem, the entropy must be constant at absolute zero. ## Notes - Entropy remains constant in a reversible process, but increases in an irreversible process. - The units of entropy are calories per Kelvin (°K) or Joules per Kelvin (J/K) - Entropy (S) of a system is a physical property like internal energy U, temperature T, which can be measured in the laboratory. - The Second Law of Thermodynamics has two important statements. The first statement is the *Kelvin-Planck statement* which states that it is not possible to construct a device that operates in a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body. The second statement is the *Clausius statement* which states that it is not possible to construct a device that operates in a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body. - The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero is zero. - The increase in entropy of an isolated system is always greater than zero. This is the *Law of Increase of Entropy*.